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A355423 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).

%I A355423 #15 Jul 02 2022 09:28:11
%S A355423 1,1,0,1,1,0,1,3,2,0,1,6,14,5,0,1,10,50,81,15,0,1,15,130,504,551,52,0,
%T A355423 1,21,280,2000,5870,4266,203,0,1,28,532,6075,35054,76872,36803,877,0,
%U A355423 1,36,924,15435,148429,684000,1111646,348543,4140,0
%N A355423 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).
%F A355423 T(0,k) = 1 and T(n,k) = Sum_{i=1..n} (Sum_{j=1..k} j^i) * binomial(n-1,i-1) * T(n-i,k) for n > 0.
%e A355423 Square array begins:
%e A355423   1,  1,    1,     1,      1,       1, ...
%e A355423   0,  1,    3,     6,     10,      15, ...
%e A355423   0,  2,   14,    50,    130,     280, ...
%e A355423   0,  5,   81,   504,   2000,    6075, ...
%e A355423   0, 15,  551,  5870,  35054,  148429, ...
%e A355423   0, 52, 4266, 76872, 684000, 4004100, ...
%Y A355423 Columns k=0-4 give: A000007, A000110, A355291, A355421, A355422.
%Y A355423 Main diagonal gives A320288.
%Y A355423 Cf. A306024, A320253.
%K A355423 nonn,tabl
%O A355423 0,8
%A A355423 _Seiichi Manyama_, Jul 01 2022