A327001 - OEIS

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A327001

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

6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..54.

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.

EXAMPLE

[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]

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 *)

CROSSREFS

A260876 (variant based on shapes).

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

Columns include: A260878, A326998.

Sequence in context: A267998 A265005 A166288 * A249138 A376555 A056648

Adjacent sequences: A326998 A326999 A327000 * A327002 A327003 A327004

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 12 2019

STATUS

approved