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%I #14 Sep 27 2022 08:46:00
%S 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,
%T 1,127,6271,25323,6556,203,64,1,1,463,129130,3086331,3068521,150349,
%U 877,128,1,1,1717,2877421,512251515,3309362716,583027547,4373461,4140,256
%N Generalized Bell numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.
%F A(n, k) = Sum_{j=0..k-1} binomial(n*k - 1, n*j) * A(n, j) for k > 0, A(n, 0) = 1.
%e [n\k][0 1 2 3 4 5 6]
%e [ - ] -----------------------------------------------------
%e [ 0 ] 1, 1, 2, 4, 8, 16, 32 A011782
%e [ 1 ] 1, 1, 2, 5, 15, 52, 203 A000110
%e [ 2 ] 1, 1, 4, 31, 379, 6556, 150349 A005046
%e [ 3 ] 1, 1, 11, 365, 25323, 3068521, 583027547 A291973
%e [ 4 ] 1, 1, 36, 6271, 3086331, 3309362716, 6626013560301 A291975
%e A260878, A326998,
%e Formatted as a triangle:
%e [1]
%e [1, 1]
%e [1, 1, 2]
%e [1, 1, 2, 4]
%e [1, 1, 4, 5, 8]
%e [1, 1, 11, 31, 15, 16]
%e [1, 1, 36, 365, 379, 52, 32]
%e [1, 1, 127, 6271, 25323, 6556, 203, 64]
%p A327001 := proc(n, k) option remember; if k = 0 then return 1 fi;
%p add(binomial(n*k - 1, n*j) * A327001(n, j), j = 0..k-1) end:
%p for n from 0 to 6 do seq(A327001(n, k), k=0..6) od; # row-wise
%t A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k-1, n*j]*A[n, j], {j, 0, k-1}]];
%t Table[A[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 27 2022 *)
%Y A260876 (variant based on shapes).
%Y Rows include: A011782, A000110, A005046, A291973, A291975.
%Y Columns include: A260878, A326998.
%Y Cf. A327000.
%K nonn,tabl
%O 0,6
%A _Peter Luschny_, Aug 12 2019
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