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A351181
a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).
2
1, 1, 17, 826, 79107, 12553011, 2979141058, 988163147091, 436562014218313, 247800100563125728, 175732698005376526429, 152264214647249387402567, 158273183995563848011907696, 194391589002961482387840145341
OFFSET
0,3
FORMULA
E.g.f.: Sum_{k>=0} (k * (exp(k*x) - 1))^k / k!.
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
MATHEMATICA
a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
PROG
(PARI) a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 04 2022
STATUS
approved