A349601 - OEIS

%I #18 Nov 25 2021 08:52:09

%S 1,1,4,32,391,6462,134974,3412187,101323674,3457536144,133333945461,

%T 5734792007584,272197255745078,14133109419794601,796883164532719216,

%U 48489515568651113516,3167153388603620859695,221021628292403019655418

%N E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)^2) - 1.

%H Seiichi Manyama, <a href="/A349601/b349601.txt">Table of n, a(n) for n = 0..361</a>

%F a(n) = Sum_{k=0..n} (2*n-k+1)^(k-1) * Stirling2(n,k).

%F 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

%t Table[Sum[(2*n - k + 1)^(k-1) * StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 25 2021 *)

%o (PARI) a(n) = sum(k=0, n, (2*n-k+1)^(k-1)*stirling(n, k, 2));

%Y Cf. A216135, A349600, A349602.

%Y Cf. A349583, A349588.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 22 2021