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%I #14 Feb 18 2022 16:11:20
%S 1,1,17,826,79107,12553011,2979141058,988163147091,436562014218313,
%T 247800100563125728,175732698005376526429,152264214647249387402567,
%U 158273183995563848011907696,194391589002961482387840145341
%N a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).
%F E.g.f.: Sum_{k>=0} (k * (exp(k*x) - 1))^k / k!.
%F a(n) ~ c * r^(2*n) * (1 + exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.942405403803582963024019065398882138211529545249588032669864757847... is the root of the equation r*(1 + exp(-1 - 1/r)) * LambertW(-exp(-1/r)/r) = -1 and c = 0.94346979328254581112250921799629823027437848684764713214690470878402... - _Vaclav Kotesovec_, Feb 18 2022
%t a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* _Amiram Eldar_, Feb 04 2022 *)
%o (PARI) a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 2));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))
%Y Cf. A108459, A229233, A229261, A282190, A308490.
%Y Cf. A350722, A351180.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 04 2022
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