A349684 - OEIS

%I #21 Nov 26 2021 09:43:41

%S 1,1,6,71,1273,30737,935217,34366971,1481055674,73255529901,

%T 4090716385913,254574063103175,17471577758796377,1310989371574276201,

%U 106774436938943155714,9381218495657924393523,884444646528793096915853,89063007333443317630241605

%N E.g.f. satisfies: log(A(x)) = 1 - exp(-x*A(x)^3).

%H Seiichi Manyama, <a href="/A349684/b349684.txt">Table of n, a(n) for n = 0..344</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (3*n+1)^(k-1) * Stirling2(n,k).

%F a(n) ~ s * n^(n-1) / (3*sqrt(1 - r*s^3) * exp(n) * r^n), where r = -LambertW(-1/3) / exp(3 + 1/LambertW(-1/3)) = 0.15501985846382288988548853891763630846... and s = exp(1 + 1/(3*LambertW(-1/3))) = 1.5865317583949486858973892879410781361... are roots of the system of equations exp(-r*s^3) + log(s) = 1, exp(r*s^3) = 3*r*s^3. - _Vaclav Kotesovec_, Nov 26 2021

%t nterms=20;Table[Sum[(-1)^(n-k)(3n+1)^(k-1)StirlingS2[n,k],{k,0,n}],{n,0,nterms-1}] (* _Paolo Xausa_, Nov 25 2021 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(3*n+1)^(k-1)*stirling(n, k, 2));

%Y Cf. A349528, A349599.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 25 2021