OFFSET
0,8
COMMENTS
M. Mureşan defined the generalized Fubini numbers as the enumerators of the k-labeled ordered p partitions of an n-set.
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.
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*<n,j> for n>0, else 1; <n,j> denotes the Eulerian numbers.
EXAMPLE
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, ...
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 08 2015
STATUS
approved