A242228 - OEIS

%I #20 Jun 19 2024 09:27:55

%S 1,17,1651,473741,300257371,355743405917,706872713310331,

%T 2182548723605418941,9894910566488309801851,

%U 63052832687428562206049117,545439670961897317869306191611,6226501736967631584015448186252541,91619831483112536750163352484302283131

%N a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).

%H Vincenzo Librandi, <a href="/A242228/b242228.txt">Table of n, a(n) for n = 1..160</a>

%F a(n) ~ c * d^n * (n!)^3 / n^2, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.15095210978787998524366903417512193343948127919...

%F E.g.f.: Sum_{k>=1} (exp(k^2*x) - 1)^k / k. - _Seiichi Manyama_, Jun 19 2024

%t Table[Sum[k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}]

%Y Cf. A000629, A220179, A229260, A244585.

%K nonn,easy

%O 1,2

%A _Vaclav Kotesovec_, May 08 2014