A232473 -id:A232473

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A232472

2-Fubini numbers.

+10
10

2, 10, 62, 466, 4142, 42610, 498542, 6541426, 95160302, 1520385010, 26468935022, 498766780786, 10114484622062, 219641848007410, 5085371491003502, 125055112347154546, 3255163896227709422, 89416052656071565810, 2584886208925055791982, 78447137202259689678706, 2493719594804686310662382

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Table of n, a(n) for n=2..22.

S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 8.

Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).

Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.

I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1 (2014).

FORMULA

Let A(x) be the g.f. A232472, B(x) the g.f. A000670, then A(x) = (1-x)*B(x) - 1. - Sergei N. Gladkovskii, Nov 29 2013

a(n) = Sum_{k>=2} T_k*k^(n-2)/2^k where T_k is the (k-1)-st triangular number (i.e., T_k = k*(k-1)/2). - Derek Orr, Jan 01 2016

a(n) = 2*A069321(n-1). - Vincenzo Librandi, Jan 03 2016, corrected by Vaclav Kotesovec, Jul 01 2018

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 01 2018

From Peter Bala, Dec 08 2020: (Start)

a(n+2) = Sum_{k = 0..n} (k+2)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+2)^n ).

a(n+2) = Sum_{k = 0..n} 2^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+2)! ).

a(n) = 2*A069321(n-1) = A000670(n) - A000670(n-1).

a(n+1)= (1/2)*Sum_{k = 0..n} binomial(n,k)*A000670(k+1) for n >= 1.

E.g.f. with offset 0: 2*exp(2*z)/(2 - exp(z))^3 = 2 + 10*z + 62*z^2/2! + 466*z^3/3! + .... (End)

EXAMPLE

G.f.: 2*x^2 + 10*x^3 + 62*x^4 + 466*x^5 + 4142*x^6 + 42610*x^7 + 498542*x^8 + ...

MAPLE

# r-Stirling numbers of second kind (e.g., A008277, A143494, A143495):

T := (n, k, r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r, i)*(i+r)^(n-r), i = 0..k-r):

# r-Bell numbers (e.g. A000110, A005493, A005494):

B := (n, r) -> add(T(n, k, r), k=r..n);

SB := r -> [seq(B(n, r), n=r..30)];

SB(2);

# r-Fubini numbers (e.g., A000670, A232472, A232473, A232474):

F := (n, r) -> add((k)!*T(n, k, r), k=r..n);

SF := r -> [seq(F(n, r), n=r..30)];

SF(2);

MATHEMATICA

Rest[max=20; t=Sum[n^(n - 1) x^n / n!, {n, 1, max}]; 2 Range[0, max]!CoefficientList[Series[D[1 / (1 - y (Exp[x] - 1)), y] /.y->1, {x, 0, max}], x]] (* Vincenzo Librandi Jan 03 2016 *)

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 2], {n, 2, 22}] (* Jean-François Alcover, Mar 30 2016 *)

PROG

(Magma) r:=2; r_Fubini:=func<n, r | &+[Factorial(k)*&+[(-1)^(k+h+r)*(h+r)^(n-r)/(Factorial(h)*Factorial(k-h-r)): h in [0..k-r]]: k in [r..n]]>;

[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

CROSSREFS

Cf. A000110, A000670, A005493, A005494, A008277, A069321, A143494, A143495, A232472, A232473, A232474.

Column k=1 of A122101.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 27 2013

STATUS

approved

A232474

4-Fubini numbers.

+10
5

24, 216, 2184, 24696, 310344, 4304376, 65444424, 1083832056, 19437971784, 375544415736, 7779464328264, 172062025581816, 4047849158698824, 100946105980181496, 2660400563437957704, 73890563849015945976, 2157336929022064219464, 66059202473570840113656, 2116993226046938197020744

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 4..200

Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).

I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.

Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

FORMULA

From Peter Bala, Dec 16 2020: (Start)

a(n+4) = Sum_{k = 0..n} (k+4)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+4)^n ).

a(n+4) = Sum_{k = 0..n} 4^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+4)! ).

E.g.f. with offset 0: 24*exp(4*z)/(2 - exp(z))^5 = 24 + 216*z + 2184*z^2/2! + 24696*z^3/3! + .... (End)

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

MAPLE

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):

T := (n, k, r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r, i)*(i+r)^(n-r), i = 0..k-r):

# r-Bell numbers (e.g. A000110, A005493, A005494):

B := (n, r) -> add(T(n, k, r), k=r..n);

SB := r -> [seq(B(n, r), n=r..30)];

SB(2);

# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):

F := (n, r) -> add((k)!*T(n, k, r), k=r..n);

SF := r -> [seq(F(n, r), n=r..30)];

SF(4);

MATHEMATICA

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 4], {n, 4, 22}] (* Jean-François Alcover, Mar 30 2016 *)

PROG

(Magma) r:=4; r_Fubini:=func<n, r | &+[Factorial(k)*&+[(-1)^(k+h+r)*(h+r)^(n-r)/(Factorial(h)*Factorial(k-h-r)): h in [0..k-r]]: k in [r..n]]>;

[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

CROSSREFS

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A232472, A232473, A232474.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 27 2013

STATUS

approved

A338875

Array T(n, m) read by ascending antidiagonals: numerators of shifted Fubini numbers F(n, m) where m >= 0.

+10
3

1, 1, 1, 3, 1, 1, 13, 5, 1, 1, 75, 2, 5, 1, 1, 541, 191, 29, 29, 1, 1, 4683, 76, 263, 149, 7, 1, 1, 47293, 5081, 4157, 24967, 2687, 727, 1, 1, 545835, 674, 93881, 115567, 44027, 66247, 631, 1, 1, 7087261, 386237, 21209, 377909, 31627, 37728769, 354061, 4481, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..54.

Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

FORMULA

T(n, m) = numerator(F(n, m)).

F(n, m) = n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, -1/(m + 2)!, ..., (-1)^(n-1)/(m + n)! (see Proposition 5.1 in Komatsu).

F(n, m) = n!*Sum_{k=0..n-1} F(k, m)/((n - k + m)!*k!) for n > 0 and m >= 0 with F(0, m) = 1 (see Lemma 5.2).

F(n, m) = [x^n] n!*x^m/(x^m - exp(x) + E_m(x)), where E_m(x) = Sum_{n=0..m} x^n/n! (see Theorem 5.3 in Komatsu).

F(n, m) = n!*Sum_{k=1..n} Sum_{i_1+...+i_k=n, i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 and m >= 0 (see Theorem 5.4).

F(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).

F(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(n-t_1-...-t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 5.7 in Komatsu).

(-1)^(n-1)/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in F(1, m), 1, 0, ..., 0 and whose first column consists in F(1, m), F(2, m)/2!, ..., F(n, m)/n! for n > 0 (see Theorem 5.8 in Komatsu).

Sum_{k=0..n} binomial(n, k)*F(k, m)*F(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! + 1)/(m*m!))^(n-l-1)*(l*(m! + 1) - m)/l!*F(l, m) - (n - m)/m*F(n, m) for m > 0 (see Theorem 5.11 in Komatsu).

EXAMPLE

Array T(n, m):

n\m| 0 1 2 3 ...

---+--------------------------------

0 | 1 1 1 1 ...

1 | 1 1 1 1 ...

2 | 3 5 5 29 ...

3 | 13 2 29 149 ...

...

Related table of shifted Fubini numbers F(n, m):

1 1 1 1 ...

1 1/2 1/6 1/24 ...

3 5/6 5/36 29/1440 ...

13 2 29/180 149/11520 ...

...

MATHEMATICA

F[n_, m_]:=n!Coefficient[Series[x^m/(x^m-Exp[x]+Sum[x^k/k!, {k, 0, m}]), {x, 0, n}], x, n]; Table[Numerator[F[n-m, m]], {n, 0, 9}, {m, 0, n}]//Flatten

PROG

(PARI) tm(n, m) = {my(m = matrix(n, n, i, j, if (i==1, if (j==1, 1/(m + 1)!, if (j==2, 1)), if (j==1, (-1)^(i+1)/(m + i)!)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }

T(n, m) = numerator(n!*matdet(tm(n, m))); \\ Michel Marcus, Dec 31 2020

CROSSREFS

Cf. A000012 (n = 0 and n = 1), A000670 (m = 0), A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.

Cf. A338876 (denominators).

KEYWORD

nonn,frac,tabl

AUTHOR

Stefano Spezia, Dec 25 2020

STATUS

approved

A338876

Array T(n, m) read by ascending antidiagonals: denominators of shifted Fubini numbers F(n, m) where m >= 0.

+10
3

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 1, 36, 24, 1, 1, 30, 180, 1440, 120, 1, 1, 3, 1080, 11520, 2400, 720, 1, 1, 42, 9072, 2419200, 2016000, 1814400, 5040, 1, 1, 1, 90720, 11612160, 60480000, 435456000, 12700800, 40320, 1, 1, 90, 7776, 33177600, 69120000, 548674560000, 21337344000, 812851200, 362880, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

FORMULA

T(n, m) = denominator(F(n, m)).

F(n, m) = n!*Sum_{k=0..n-1} F(k, m)/((n - k + m)!*k!) for n > 0 and m >= 0 with F(0, m) = 1 (see Lemma 5.2).

F(n, m) = [x^n] n!*x^m/(x^m - exp(x) + E_m(x)), where E_m(x) = Sum_{n=0..m} x^n/n! (see Theorem 5.3 in Komatsu).

F(n, m) = n!*Sum_{k=1..n} Sum_{i_1+...+i_k=n, i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 and m >= 0 (see Theorem 5.4).

F(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).

F(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(n-t_1-...-t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 5.7 in Komatsu).

Sum_{k=0..n} binomial(n, k)*F(k, m)*F(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! + 1)/(m*m!))^(n-l-1)*(l*(m! + 1) - m)/l!*F(l, m) - (n - m)/m*F(n, m) for m > 0 (see Theorem 5.11 in Komatsu).

EXAMPLE

Array T(n, m):

n\m| 0 1 2 3 ...

---+--------------------------------

0 | 1 1 1 1 ...

1 | 1 2 6 24 ...

2 | 1 6 36 1440 ...

3 | 1 1 180 11520 ...

...

Related table of shifted Fubini numbers F(n, m):

1 1 1 1 ...

1 1/2 1/6 1/24 ...

3 5/6 5/36 29/1440 ...

13 2 29/180 149/11520 ...

...

MATHEMATICA

F[n_, m_]:=n!Coefficient[Series[x^m/(x^m-Exp[x]+Sum[x^k/k!, {k, 0, m}]), {x, 0, n}], x, n]; Table[Denominator[F[n-m, m]], {n, 0, 9}, {m, 0, n}]//Flatten

PROG

(PARI) tm(n, m) = {my(m = matrix(n, n, i, j, if (i==1, if (j==1, 1/(m + 1)!, if (j==2, 1)), if (j==1, (-1)^(i+1)/(m + i)!)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }

T(n, m) = denominator(n!*matdet(tm(n, m))); \\ Michel Marcus, Dec 31 2020

CROSSREFS

Cf. A000012 (n = 0 or m = 0), A000142, A000670, A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.

Cf. A338875 (numerators).

KEYWORD

nonn,frac,tabl

AUTHOR

Stefano Spezia, Dec 25 2020

STATUS

approved

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