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A184640
Number of partitions of n having no parts with multiplicity 5.
8
1, 1, 2, 3, 5, 6, 11, 14, 21, 28, 39, 51, 72, 92, 124, 160, 210, 266, 349, 438, 562, 704, 892, 1107, 1395, 1720, 2141, 2631, 3249, 3965, 4873, 5916, 7216, 8730, 10585, 12742, 15387, 18443, 22151, 26466, 31646, 37659, 44873, 53212, 63149, 74666, 88295
OFFSET
0,3
LINKS
FORMULA
a(n) = A000041(n) - A183562(n).
a(n) = A183568(n,0) - A183568(n,5).
G.f.: Product_{j>0} (1-x^(5*j)+x^(6*j))/(1-x^j).
EXAMPLE
a(5) = 6, because 6 partitions of 5 have no parts with multiplicity 5: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 5, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 18 2011
STATUS
approved