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a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(k*n) * Stirling1(n,k).
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%I #17 Feb 04 2022 08:22:55

%S 1,1,33,118484,103098352618,35763050751038414134,

%T 7426387531294394110580641088438,

%U 1294894837982331434068068403253026516109577144,253092742000650212462862632240661689524832716838851180353875064

%N a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(k*n) * Stirling1(n,k).

%H Seiichi Manyama, <a href="/A351138/b351138.txt">Table of n, a(n) for n = 0..26</a>

%F E.g.f.: Sum_{k>=0} (-log(1 - k^k*x))^k.

%F a(n) ~ n! * n^(n^2). - _Vaclav Kotesovec_, Feb 03 2022

%t a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(k*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 9, 0] (* _Amiram Eldar_, Feb 02 2022 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(k*n)*stirling(n, k, 1));

%o (PARI) my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^k*x))^k)))

%Y Cf. A007840, A320096, A351136, A351137.

%Y Cf. A249584, A351135.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 02 2022