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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.

%I A257565 #30 Nov 25 2021 12:34:14
%S A257565 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,
%T A257565 4683,4501,1015,121,11,1,1,1,47293,66605,17641,2169,181,13,1,1,1,
%U A257565 545835,1149877,367927,48601,3971,253,15,1,1,1,7087261,22687565,8952553,1306809,108901,6565,337,17,1,1
%N 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.
%C A257565 M. Mureşan defined the generalized Fubini numbers as the enumerators of the k-labeled ordered p partitions of an n-set.
%D A257565 M. Mureşan, On the generalized Fubini numbers. (Romanian) Stud. Cercet. Mat. 37, 70-76 (1985).
%H A257565 Robert Gill, <a href="https://doi.org/10.1016/S0012-365X(97)00187-8">The number of elements in a generalized partition semilattice</a>, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
%H A257565 N. Kilar and Y. Simsek, <a href="https://doi.org/10.4134/JKMS.j160597">A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials</a>, J. Korean Math. Soc. 54 (5) (2017) 1605-1621.
%F A257565 E.g.f. of column k: 1+1/(1/(exp(z)-1)-k).
%F A257565 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.
%F A257565 A(n,k) = Sum_{j=0..n-1} k^(n-j-1)*(k+1)^j*<n,j> for n>0, else 1; <n,j> denotes the Eulerian numbers.
%e A257565       1,       1,       1,       1,        1,         1, ...  A000012
%e A257565       1,       1,       1,       1,        1,         1, ...  A000012
%e A257565       1,       3,       5,       7,        9,        11, ...  A005408
%e A257565       1,      13,      37,      73,      121,       181, ...  A003154
%e A257565       1,      75,     365,    1015,     2169,      3971, ...  A193252
%e A257565       1,     541,    4501,   17641,    48601,    108901, ...
%e A257565       1,    4683,   66605,  367927,  1306809,   3583811, ...
%e A257565       1,   47293, 1149877, 8952553, 40994521, 137595781, ...
%e A257565 A000012, A000670, A050351, A050352,  A050353,
%p A257565 F := proc(n,k) option remember; 1+k*add(binomial(n,j)*F(j,k),j=1..n-1) end:
%p A257565 seq(print(seq(F(n-k,k),k=0..n)), n=0..7); # triangular form
%p A257565 egf := k -> 1+1/(1/(exp(z)-1)-k): # egf of column k
%p A257565 for k from 0 to 4 do seq(j!*coeff(series(egf(k),z,10),z,j),j=0..8) od;
%p A257565 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);
%t A257565 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 *)
%Y A257565 Rows: A005408, A003154, A193252.
%Y A257565 Columns: A000670, A050351, A050352, A050353.
%K A257565 nonn,tabl
%O A257565 0,8
%A A257565 _Peter Luschny_, May 08 2015