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%I #31 Feb 04 2022 09:51:26
%S 1,1,33,118483,103098350565,35763050750177408011,
%T 7426387531294259002278007386693,
%U 1294894837982331421844458945612619053737859003,253092742000650212461957357208907985560332648454746968725711765
%N a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling2(n,k).
%H Seiichi Manyama, <a href="/A351117/b351117.txt">Table of n, a(n) for n = 0..26</a>
%F E.g.f.: Sum_{k>=0} (exp(k^k*x) - 1)^k.
%F G.f.: Sum_{k>=0} k! * (k^k*x)^k/Product_{j=1..k} (1 - k^k*j*x).
%F a(n) ~ n! * n^(n^2). - _Vaclav Kotesovec_, Feb 04 2022
%p a:= n-> add(k!*k^(k*n)*Stirling2(n,k), k=0..n):
%p seq(a(n), n=0..10); # _Alois P. Heinz_, Feb 01 2022
%t a[0] = 1; a[n_] := Sum[k! * k^(k*n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 9, 0] (* _Amiram Eldar_, Feb 02 2022 *)
%o (PARI) a(n) = sum(k=0, n, k!*k^(k*n)*stirling(n, k, 2));
%o (PARI) my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^k*x)-1)^k)))
%o (PARI) my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^k*x)^k/prod(j=1, k, 1-k^k*j*x)))
%Y Cf. A000670, A122399, A229260, A242229.
%Y Cf. A249584.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 01 2022
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