cp's OEIS Frontend

A349557 E.g.f. satisfies: log(A(x)) = (exp(x*A(x)) - 1) * A(x).

%I A349557 #15 Jul 15 2022 21:49:07
%S A349557 1,1,6,68,1163,26787,778128,27325321,1126308870,53323302708,
%T A349557 2851990661789,170088808988705,11192134680722586,805521092432042573,
%U A349557 62950026461699015998,5308512876799649771192,480492707646769163920059,46464318322169305448661915
%N A349557 E.g.f. satisfies: log(A(x)) = (exp(x*A(x)) - 1) * A(x).
%H A349557 Alois P. Heinz, <a href="/A349557/b349557.txt">Table of n, a(n) for n = 0..345</a>
%F A349557 a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * Stirling2(n,k).
%F A349557 a(n) ~ sqrt(s^3 * (1+s) / (1 + r^2*s^2*(1+s) + r*s*(3 + 2*s))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1609673785833512641321517974482987852086944930869... and s = 1.597727491873940099115048788232158935283220293884... are real roots of the system of equations exp(r*s)*s = s + log(s), exp(r*s)*(1 + r*s) = 1 + 1/s. - _Vaclav Kotesovec_, Nov 22 2021
%t A349557 a[n_] := Sum[(n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 22 2021 *)
%o A349557 (PARI) a(n) = sum(k=0, n, (n+k+1)^(k-1)*stirling(n, k, 2));
%Y A349557 Cf. A052880, A349558, A349560.
%K A349557 nonn
%O A349557 0,3
%A A349557 _Seiichi Manyama_, Nov 21 2021