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
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
Programs
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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
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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.
Comments