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%I #6 Nov 16 2017 17:28:45
%S 1,1,0,1,0,0,1,0,1,0,1,0,2,2,0,1,0,3,4,9,0,1,0,4,6,24,44,0,1,0,5,8,45,
%T 128,265,0,1,0,6,10,72,252,880,1854,0,1,0,7,12,105,416,1935,6816,
%U 14833,0,1,0,8,14,144,620,3520,16146,60032,133496,0,1,0,9,16,189,864,5725,31104,153657,589312,1334961,0
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x)^k.
%C A(n,k) is the k-fold exponential convolution of A000166 with themselves, evaluated at n.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F E.g.f. of column k: exp(-k*x)/(1 - x)^k.
%e E.g.f. of column k: A_k(x) = 1 + k*x^2/2! + 2*k*x^3/3! + 3*k*(k + 2)*x^4/4! + 4*k*(5*k + 6)*x^5/5! + 5*k*(3*k^2 + 26*k + 24)*x^6/6! + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 0, 0, 0, 0, 0, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 2, 4, 6, 8, 10, ...
%e 0, 9, 24, 45, 72, 105, ...
%e 0, 44, 128, 252, 416, 620, ...
%t Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0..3 give A000007, A000166, A087981, A137775.
%Y Rows n=0..3 give A000012, A000004, A001477, A005843.
%Y Main diagonal gives A295182.
%Y Cf. A008279, A265609.
%K nonn,tabl
%O 0,13
%A _Ilya Gutkovskiy_, Nov 16 2017