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

0,8

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

Table of n, a(n) for n=0..65.

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, ...
A000012, A000670, A050351, A050352,  A050353,

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

Rows: A005408, A003154, A193252.

Columns: A000670, A050351, A050352, A050353.

Sequence in context: A121585 A261959 A348988 * A276121 A262809 A331568

Adjacent sequences: A257562 A257563 A257564 * A257566 A257567 A257568

Keyword

nonn,tabl

Author

Peter Luschny, May 08 2015

Status

approved