A327001 Generalized Bell numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 4, 5, 8, 1, 1, 11, 31, 15, 16, 1, 1, 36, 365, 379, 52, 32, 1, 1, 127, 6271, 25323, 6556, 203, 64, 1, 1, 463, 129130, 3086331, 3068521, 150349, 877, 128, 1, 1, 1717, 2877421, 512251515, 3309362716, 583027547, 4373461, 4140, 256

Offset: 0

Author

Peter Luschny, Aug 12 2019

Keywords

• nonn
• tabl

Examples

			[n\k][0  1   2        3        4           5             6]
[ - ] -----------------------------------------------------
[ 0 ] 1, 1,  2,       4,       8,         16,            32  A011782
[ 1 ] 1, 1,  2,       5,      15,         52,           203  A000110
[ 2 ] 1, 1,  4,      31,     379,       6556,        150349  A005046
[ 3 ] 1, 1, 11,     365,   25323,    3068521,     583027547  A291973
[ 4 ] 1, 1, 36,    6271, 3086331, 3309362716, 6626013560301  A291975
       A260878, A326998,
Formatted as a triangle:
[1]
[1, 1]
[1, 1,   2]
[1, 1,   2,    4]
[1, 1,   4,    5,     8]
[1, 1,  11,   31,    15,    16]
[1, 1,  36,  365,   379,   52,  32]
[1, 1, 127, 6271, 25323, 6556, 203, 64]
		

Crossrefs

A260876 (variant based on shapes).

Rows include: A011782, A000110, A005046, A291973, A291975.

Columns include: A260878, A326998.

Cf. A327000.

Programs

Maple

A327001 := proc(n, k) option remember; if k = 0 then return 1 fi;
add(binomial(n*k - 1, n*j) * A327001(n, j), j = 0..k-1) end:
for n from 0 to 6 do seq(A327001(n, k), k=0..6) od; # row-wise

Mathematica

A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k-1, n*j]*A[n, j], {j, 0, k-1}]];
Table[A[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 27 2022 *)

Formula

A(n, k) = Sum_{j=0..k-1} binomial(n*k - 1, n*j) * A(n, j) for k > 0, A(n, 0) = 1.