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A295181 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.

%I A295181 #19 Apr 25 2025 16:00:32
%S A295181 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 A295181 128,265,0,1,0,6,10,72,252,880,1854,0,1,0,7,12,105,416,1935,6816,
%U A295181 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 A295181 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 A295181 A(n,k) is the k-fold exponential convolution of A000166 with themselves, evaluated at n.
%H A295181 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A295181 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A295181 E.g.f. of column k: exp(-k*x)/(1 - x)^k.
%F A295181 From _Seiichi Manyama_, Apr 25 2025: (Start)
%F A295181 A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k-1,j)/(n-j)!.
%F A295181 A(0,k) = 1, A(1,k) = 0; A(n,k) = (n-1) * (A(n-1,k) + k*A(n-2,k)). (End)
%e A295181 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 A295181 Square array begins:
%e A295181   1,   1,    1,    1,    1,    1,  ...
%e A295181   0,   0,    0,    0,    0,    0,  ...
%e A295181   0,   1,    2,    3,    4,    5,  ...
%e A295181   0,   2,    4,    6,    8,   10,  ...
%e A295181   0,   9,   24,   45,   72,  105,  ...
%e A295181   0,  44,  128,  252,  416,  620,  ...
%t A295181 Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A295181 (PARI) a(n, k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k-1, j)/(n-j)!); \\ _Seiichi Manyama_, Apr 25 2025
%Y A295181 Columns k=0..5 give A000007, A000166, A087981, A137775, A383344, A383384.
%Y A295181 Rows n=0..3 give A000012, A000004, A001477, A005843.
%Y A295181 Main diagonal gives A295182.
%Y A295181 Cf. A008279, A265609.
%K A295181 nonn,tabl
%O A295181 0,13
%A A295181 _Ilya Gutkovskiy_, Nov 16 2017