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A036035
Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.
+10
37
1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, 960, 1440, 2160
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".
Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)
Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)
Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.
The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).
Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).
When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - Alford Arnold, Jul 31 2003
A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - Alford Arnold, Sep 29 2005
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).
LINKS
Peter Luschny, Rows n = 0..25, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
John Baez, What happens when a particle gets created?
OEIS Wiki, Prime signature.
EXAMPLE
1;
2;
4, 6;
8, 12, 30;
16, 24, 36, 60, 210;
32, 48, 72, 120, 180, 420, 2310;
64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;
128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;
MAPLE
with(combinat):
A036035_row := proc(n) local e, w; w := proc(e) local i, p;
p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];
mul(p[i]^e[i], i=1..nops(e)) end:
seq(w(conjpart(e)), e = partition(n)) end:
seq(A036035_row(i), i=0..10); # Peter Luschny, Aug 01 2013
MATHEMATICA
nmax = 52; primeSignature[n_] := Sort[ FactorInteger[n], #1[[2]] > #2[[2]] & ][[All, 2]]; ip[n_] := Reverse[ Sort[#]] & /@ Split[ Sort[ IntegerPartitions[n], Length[#1] < Length[#2] & ], Length[#1] == Length[#2] & ]; tip = Flatten[ Table[ip[n], {n, 0, 8}], 2]; a[n_] := (sig = tip[[n+1]]; k = 1; While[sig =!= primeSignature[k++]]; k-1); a[0] = 1; a[1] = 2; Table[an = a[n]; Print[an]; an, {n, 0, nmax}](* Jean-François Alcover, Nov 16 2011 *)
PROG
(PARI) Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ Andrew Howroyd, Oct 19 2020
CROSSREFS
A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.
Cf. A025487, A059901, A096443.
KEYWORD
nonn,easy,nice,tabf,look
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Alford Arnold; corrected Sep 10 2002
More terms from Ray Chandler, Jul 13 2003
Definition corrected by Daniel Forgues, Jan 16 2011
STATUS
approved
A346426
Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+10
24
1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).
EXAMPLE
A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
2, 4, 11, 36, 135, 566, 2610, 13082, 70631, ...
3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, ...
5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, ...
7, 19, 64, 250, 1098, 5317, 28009, 158926, 963913, ...
11, 30, 105, 426, 1940, 9722, 52902, 309546, 1933171, ...
15, 45, 165, 696, 3281, 16972, 95129, 572402, 3670878, ...
22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
...
MAPLE
s:= proc(n) option remember; expand(`if`(n=0, 1,
x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 16 2021
STATUS
approved
A035310
Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.
+10
21
1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Ways of partitioning an n-multiset with multiplicities some partition of n.
Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..50
EXAMPLE
a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.
From Gus Wiseman, Dec 31 2019: (Start)
The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:
{{1}} {{1,1}} {{1,1,1}}
{{1,2}} {{1,1,2}}
{{1},{1}} {{1,2,3}}
{{1},{2}} {{1},{1,1}}
{{1},{1,2}}
{{1},{2,3}}
{{2},{1,1}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{3}}
(End)
MAPLE
with(numtheory):
g:= proc(n, k) option remember;
`if`(n>k, 0, 1) +`if`(isprime(n), 0,
add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))
end:
b:= proc(n, i, l)
`if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),
`if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
end:
a:= n-> b(n$2, []):
seq(a(n), n=1..10); # Alois P. Heinz, May 26 2013
MATHEMATICA
g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)
PROG
(Python)
from sympy.core.cache import cacheit
from sympy import divisors, isprime, prime
from operator import mul
@cacheit
def g(n, k):
return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))
@cacheit
def b(n, i, l):
if n==0:
p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))
return g(p, p)
else:
return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])
def a(n):
return b(n, n, [])
for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p, n)); s} \\ Andrew Howroyd, Dec 30 2020
CROSSREFS
Cf. A025487, A000041, A000110, A035098, A080688.
Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.
Cf. A035341, A093936, A095705.
Row sums of A317449.
The uniform case is A317584.
The case with empty intersection is A317755.
The strict case is A317775.
The constant case is A047968.
The set-system case is A318402.
The case of strict parts is A330783.
Multiset partitions of integer partitions are A001970.
Unlabeled multiset partitions are A007716.
Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.
KEYWORD
nonn,nice
AUTHOR
Alford Arnold
EXTENSIONS
More terms from Erich Friedman.
81199 from Alford Arnold, Mar 04 2008
a(10) from Alford Arnold, Mar 31 2008
a(10) corrected by Alford Arnold, Aug 07 2008
a(11)-a(13) from Alois P. Heinz, May 26 2013
a(14) from Alois P. Heinz, Sep 27 2014
a(15) from Alois P. Heinz, Jan 10 2015
Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved
A346500
Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+10
15
1, 1, 1, 2, 2, 2, 5, 4, 4, 5, 15, 11, 9, 11, 15, 52, 36, 26, 26, 36, 52, 203, 135, 92, 66, 92, 135, 203, 877, 566, 371, 249, 249, 371, 566, 877, 4140, 2610, 1663, 1075, 712, 1075, 1663, 2610, 4140, 21147, 13082, 8155, 5133, 3274, 3274, 5133, 8155, 13082, 21147
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i); A(2,2) = 9: 2*2*3*3, 3*3*4, 6*6, 2*3*6, 4*9, 2*2*9, 3*12, 2*18, 36.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
A(n,k) = A001055(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322765(abs(n-k),min(n,k)).
EXAMPLE
A(2,2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, ...
1, 2, 4, 11, 36, 135, 566, 2610, ...
2, 4, 9, 26, 92, 371, 1663, 8155, ...
5, 11, 26, 66, 249, 1075, 5133, 26683, ...
15, 36, 92, 249, 712, 3274, 16601, 91226, ...
52, 135, 371, 1075, 3274, 10457, 56135, 325269, ...
203, 566, 1663, 5133, 16601, 56135, 198091, 1207433, ...
877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
...
MAPLE
g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
g(n/d, d)), d=divisors(n) minus {1, n}))
end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n-1, j-1), j=1..n))
end:
A:= proc(n, k) option remember; `if`(n<k, A(k, n),
`if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
*binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz's second program *)
CROSSREFS
Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
Main diagonal gives A020555.
First upper (or lower) diagonal gives A322766.
Second upper (or lower) diagonal gives A322767.
Antidiagonal sums give A346490.
A(2n,n) gives A322769.
Cf. A001055, A002110, A322765, A346426, A346517.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 20 2021
STATUS
approved
A096443
Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).
+10
14
1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 11, 15, 7, 12, 16, 21, 26, 36, 52, 11, 19, 29, 38, 31, 52, 74, 66, 92, 135, 203, 15, 30, 47, 64, 57, 98, 141, 109, 137, 198, 296, 249, 371, 566, 877, 22, 45, 77, 105, 97, 171, 250, 109, 212, 269, 392, 592, 300, 444, 560, 850, 1315, 712, 1075
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The signature of a multiset is the partition consisting of the multiplicities of its elements; e.g., {a,a,a,b,c} is represented by [3,1,1]. The Mathematica order for partitions orders by ascending number of total elements, then by descending numerical order of its representation. The list begins:
n.....#elements.....n-th partition
0.....0 elements:....[]
1.....1 element:.....[1]
2.....2 elements:....[2]
3....................[1,1]
4.....3 elements:....[3]
5....................[2,1]
6....................[1,1,1]
7.....4 elements:....[4]
8....................[3,1]
9....................[2,2]
10...................[2,1,1]
11...................[1,1,1,1]
12....5 elements:....[5]
13...................[4,1]
A000041 and A000110 are subsequences for conjugate partitions. A000070 and A035098 are also subsequences for conjugate partitions. - Alford Arnold, Dec 31 2005
A002774 and A020555 is another pair of subsequences for conjugate partitions. - Franklin T. Adams-Watters, May 16 2006
LINKS
Table of n, a(n) for n=0..63.
Jun Kyo Kim and Sang Guen Hahn, Recursive Formulae for the Multiplicative Partition Function, Internat. J. Math. & Math. Sci., 22(1) (1999), 213-216.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See P(n) page 3.
EXAMPLE
The 10th partition is [2,1,1]. The partitions of a multiset whose elements have multiplicities 2,1,1 - for example, {a,a,b,c} - are:
{{a,a,b,c}}
{{a,a,b},{c}}
{{a,a,c},{b}}
{{a,b,c},{a}}
{{a,a},{b,c}}
{{a,b},{a,c}}
{{a,a},{b},{c}}
{{a,b},{a},{c}}
{{a,c},{a},{b}}
{{b,c},{a},{a}}
{{a},{a},{b},{c}}
We see there are 11 partitions of this multiset, so a(10)=11.
Also, a(n) is the number of distinct factorizations of A063008(n). For example, A063008(10) = 60 and 60 has 11 factorizations: 60, 30*2, 20*3, 15*4, 15*2*2, 12*5, 10*6, 10*3*2, 6*5*2, 5*4*3, 5*3*2*2 which confirms that a(10) = 11.
MATHEMATICA
MultiPartiteP[n : {___Integer?NonNegative}] :=
Block[{p, $RecursionLimit = 1024, firstPositive},
firstPositive =
Compile[{{vv, _Integer, 1}},
Module[{k = 1}, Do[If[el == 0, k++, Break[]], {el, vv}]; k]];
p[{0 ...}] := 1;
p[v_] :=
p[v] = Module[{len = Length[v], it, k, zeros, sum, pos, gcd},
it = Array[k, len];
pos = firstPositive[v];
zeros = ConstantArray[0, len];
sum = 0;
Do[If[it == zeros, Continue[]];
gcd = GCD @@ it;
sum += it[[pos]] DivisorSigma[-1, gcd] p[v - it]; ,
Evaluate[Sequence @@ Thread[{it, 0, v}]]];
sum/v[[pos]]];
p[n]];
ParallelMap[MultiPartiteP,
Flatten[Table[IntegerPartitions[k], {k, 0, 8}], 1]]
(* Oleksandr Pavlyk, Jan 23 2011 *)
CROSSREFS
Cf. A035098, A035310.
KEYWORD
nonn
AUTHOR
Jon Wild, Aug 11 2004
EXTENSIONS
Edited by Franklin T. Adams-Watters, May 16 2006
STATUS
approved
A035341
Sum of ordered factorizations over all prime signatures with n factors.
+10
12
1, 1, 5, 25, 173, 1297, 12225, 124997, 1492765, 19452389, 284145077, 4500039733, 78159312233, 1460072616929, 29459406350773, 634783708448137, 14613962109584749, 356957383060502945, 9241222160142506097, 252390723655315856437, 7260629936987794508973
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.
When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.
Row sums of A095705. - David Wasserman, Feb 22 2008
From Ludovic Schwob, Sep 23 2023: (Start)
a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:
[1 1 1] [1 2] [2 1] [3]
.
[1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]
[1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]
.
[1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
[1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
[1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
.
[0 1 0] [0 1 0] [0 0 1] [0 0 1]
[1 0 0] [0 0 1] [1 0 0] [0 1 0]
[0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 36 terms from David Wasserman)
Eric Weisstein's World of Mathematics, Perfect Partition
FORMULA
a(n) ~ c * n! / log(2)^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.8246323... . - Vaclav Kotesovec, Jan 21 2017
EXAMPLE
a(3) = 25 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=4, f(12)=8, f(30)=13 and 4+8+13 = 25.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 29 2015
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*If[j == 0, 1, Binomial[i + k - 1, k - 1]^j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz, updated Dec 15 2020 *)
PROG
(PARI)
R(n, k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1, j)*x^j + O(x*x^n)))
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023
CROSSREFS
Cf. A005651, A025487, A035310, A255906, A365961.
Row sums of A261719.
KEYWORD
nonn,nice
AUTHOR
Alford Arnold
EXTENSIONS
More terms from Erich Friedman.
More terms from David Wasserman, Feb 22 2008
STATUS
approved
A126442
Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.
+10
10
1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 7, 12, 21, 36, 52, 11, 19, 38, 74, 135, 203, 15, 30, 64, 141, 296, 566, 877, 22, 45, 105, 250, 592, 1315, 2610, 4140, 30, 67, 165, 426, 1098, 2752, 6393, 13082, 21147, 42, 97, 254, 696, 1940, 5317, 13960, 33645, 70631, 115975
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
First in a series of triangular arrays which comprise subsequences of A096443(n).
The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011
Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014
LINKS
Table of n, a(n) for n=1..55.
Tilman Piesk, Partitions of multisets (Wikiversity)
EXAMPLE
This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
The array begins:
1
2 2
3 4 5
5 7 11 15
7 12 21 36 52
MATHEMATICA
(* The triangle is flattened to a sequence. *)
t[n_, k_] := Sum[StirlingS2[n, j] * Binomial[-1 + i + j, i] * PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Table[ t[n, k], {k, 10}, {n, 0, k - 1}] // Flatten (* George Beck, Jan 10 2011 *)
CROSSREFS
Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.
KEYWORD
nonn,tabl
AUTHOR
Alford Arnold, Jan 28 2007
EXTENSIONS
Definition clarified by George Beck, Jan 11 2011
STATUS
approved
A322765
Array read by upwards antidiagonals: T(m,n) = number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.
+10
9
1, 1, 2, 2, 4, 9, 5, 11, 26, 66, 15, 36, 92, 249, 712, 52, 135, 371, 1075, 3274, 10457, 203, 566, 1663, 5133, 16601, 56135, 198091, 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, 4140, 13082, 43263, 149410, 537813, 2014321, 7837862, 31638625, 132315780
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).
From Alois P. Heinz, Jul 21 2021: (Start)
A(n,k) = A001055(A002110(n+k)*A002110(k)).
A(n,k) = A346500(n+k,k). (End)
EXAMPLE
The array begins:
1, 2, 9, 66, 712, 10457, 198091, ...
1, 4, 26, 249, 3274, 56135, 1207433, ...
2, 11, 92, 1075, 16601, 325269, 7837862, ...
5, 36, 371, 5133, 91226, 2014321, 53840640, ...
15, 135, 1663, 26683, 537813, 13241402, 389498179, ...
52, 566, 8155, 149410, 3376696, 91914202, 2955909119, ...
203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
...
MAPLE
B := n -> combinat[bell](n):
P := proc(m, n) local k; global B; option remember;
if n = 0 then B(m) else
(1/2)*( P(m+2, n-1) + P(m+1, n-1) + add( binomial(n-1, k)*P(m, k), k=0..n-1) ); fi; end; # P(m, n) (which is Knuth's notation) is T(m, n)
MATHEMATICA
P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)
PROG
(PARI) {T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2)} \\ Seiichi Manyama, Nov 21 2020
CROSSREFS
Rows include A020555, A322766, A322767.
Columns include A000110, A035098, A322764, A322768.
Main diagonal is A322769.
See A322770 for partitions into distinct parts.
Cf. A001055, A002110, A346500.
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Dec 30 2018
STATUS
approved
A059606
Expansion of (1/2)*(exp(2*x)-1)*exp(exp(x)-1).
+10
8
0, 1, 4, 16, 68, 311, 1530, 8065, 45344, 270724, 1709526, 11376135, 79520644, 582207393, 4453142140, 35500884556, 294365897104, 2533900264547, 22604669612078, 208656457858161, 1990060882027600
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Starting (1, 4, 16, 68, 311, ...), = A008277 * A000217, i.e., the product of the Stirling2 triangle and triangular series. - Gary W. Adamson, Jan 31 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladeta Jovovic, More information.
FORMULA
a(n) = Sum_{i=0..n} Stirling2(n, i)*binomial(i+1, 2).
a(n) = (1/2)*(Bell(n+2)-Bell(n+1)-Bell(n)). - Vladeta Jovovic, Sep 23 2003
G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018
a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
MAPLE
s := series(1/2*(exp(2*x)-1)*exp(exp(x)-1), x, 21): for i from 0 to 20 do printf(`%d, `, i!*coeff(s, x, i)) od:
MATHEMATICA
With[{nn=20}, CoefficientList[Series[((Exp[2x]-1)Exp[Exp[x]-1])/2, {x, 0, nn}] , x] Range[0, nn]!] (* Harvey P. Dale, Nov 10 2011 *)
CROSSREFS
Cf. A000110, A005493, A059604, A059605.
Cf. A035098.
Cf. A008277, A000217.
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Jan 29 2001
STATUS
approved
A087648
a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).
+10
7
1, 3, 9, 31, 120, 514, 2407, 12205, 66491, 386699, 2388096, 15589732, 107165081, 773106715, 5836100685, 45981026703, 377230766908, 3215977070706, 28437411817135, 260380616093533, 2464930698184351, 24091925888687459, 242802079705721156, 2520198597834860148
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - Franklin T. Adams-Watters, Jun 07 2006
Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets. Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - Alois P. Heinz, Jul 30 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
MATHEMATICA
f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* Zerinvary Lajos, Mar 31 2007 *)
(#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0, 30]], 3, 1] (* Harvey P. Dale, Jul 20 2021 *)
PROG
(Magma) [(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011
CROSSREFS
Cf. A000110, A035098, A059606.
Main diagonal of A120057, row sums of A120095.
Column 1 of array in A322770.
Row n=2 of A346520.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 23 2003
STATUS
approved
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