A349601 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)^2) - 1.
1, 1, 4, 32, 391, 6462, 134974, 3412187, 101323674, 3457536144, 133333945461, 5734792007584, 272197255745078, 14133109419794601, 796883164532719216, 48489515568651113516, 3167153388603620859695, 221021628292403019655418
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..361
Programs
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Mathematica
Table[Sum[(2*n - k + 1)^(k-1) * StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 25 2021 *) -
PARI
a(n) = sum(k=0, n, (2*n-k+1)^(k-1)*stirling(n, k, 2));
Formula
a(n) = Sum_{k=0..n} (2*n-k+1)^(k-1) * Stirling2(n,k).
a(n) ~ s * n^(n-1) / (sqrt(2*(1 + 2*r*s^2) - 2/(1 + log(s))) * exp(n) * r^n), where r = 0.2229533052706631261980294005821031136702825459439... and s = 1.759796045489338472919926226485178994146849909897... are roots of the system of equations exp(r*s^2) = 1 + s*log(s), 2*exp(r*s^2)*r*s = 1 + log(s). - Vaclav Kotesovec, Nov 25 2021