A257565 Generalized Fubini numbers. Square array read by ascending antidiagonals, A(n,k) = 1 + k*(Sum_{j=1..n-1} C(n,j)*A(j,k)); n>=0 and k>=0.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 5, 1, 1, 1, 75, 37, 7, 1, 1, 1, 541, 365, 73, 9, 1, 1, 1, 4683, 4501, 1015, 121, 11, 1, 1, 1, 47293, 66605, 17641, 2169, 181, 13, 1, 1, 1, 545835, 1149877, 367927, 48601, 3971, 253, 15, 1, 1, 1, 7087261, 22687565, 8952553, 1306809, 108901, 6565, 337, 17, 1, 1

Offset: 0

Author

Peter Luschny, May 08 2015

Keywords

• nonn
• tabl

Comments

M. Mureşan defined the generalized Fubini numbers as the enumerators of the k-labeled ordered p partitions of an n-set.

Examples

			      1,       1,       1,       1,        1,         1, ...  A000012
      1,       1,       1,       1,        1,         1, ...  A000012
      1,       3,       5,       7,        9,        11, ...  A005408
      1,      13,      37,      73,      121,       181, ...  A003154
      1,      75,     365,    1015,     2169,      3971, ...  A193252
      1,     541,    4501,   17641,    48601,    108901, ...
      1,    4683,   66605,  367927,  1306809,   3583811, ...
      1,   47293, 1149877, 8952553, 40994521, 137595781, ...
A000012, A000670, A050351, A050352,  A050353,
		

References

• M. Mureşan, On the generalized Fubini numbers. (Romanian) Stud. Cercet. Mat. 37, 70-76 (1985).

Links

• Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
• N. Kilar and Y. Simsek, A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials, J. Korean Math. Soc. 54 (5) (2017) 1605-1621.

Crossrefs

Rows: A005408, A003154, A193252.

Columns: A000670, A050351, A050352, A050353.

Programs

Maple

F := proc(n,k) option remember; 1+k*add(binomial(n,j)*F(j,k),j=1..n-1) end:
seq(print(seq(F(n-k,k),k=0..n)), n=0..7); # triangular form
egf := k -> 1+1/(1/(exp(z)-1)-k): # egf of column k
for k from 0 to 4 do seq(j!*coeff(series(egf(k),z,10),z,j),j=0..8) od;
A := (n,k) -> `if`(n=0,1,add(k^(n-j-1)*(k+1)^j*combinat:-eulerian1(n,j),j=0..n-1)): seq(print(seq(A(n,k),k=0..5)),n=0..7);

Mathematica

A[n_, k_] := A[n, k] = 1 + k Sum[Binomial[n, j] A[j, k], {j, 1, n - 1}]; Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2016 *)

Formula

E.g.f. of column k: 1+1/(1/(exp(z)-1)-k).

A(n,k) = Sum_{j=0..n-1} k^j*j!*{n,j+1} for n>0, else 1; {n,j} denotes the Stirling subset numbers.

A(n,k) = Sum_{j=0..n-1} k^(n-j-1)*(k+1)^j* for n>0, else 1; denotes the Eulerian numbers.