cp's OEIS Frontend

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

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