A357650 - OEIS

%I #12 Oct 07 2022 15:46:57

%S 1,0,1,12,113,1000,8977,86996,959905,12303888,179038689,2840696540,

%T 47684181393,835731314808,15277172343409,292597596283684,

%U 5900038421042753,125488177929542944,2809541905807203009,65903118624174027436,1610968753088423886257

%N Expansion of e.g.f. cosh( (exp(4*x) - 1)/4 ).

%F a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * Stirling2(n,2*k).

%F a(n) ~ 2^(2*n-1) * exp(n/LambertW(4*n) - n - 1/4) * n^n / (LambertW(4*n)^n * sqrt(1 + LambertW(4*n))). - _Vaclav Kotesovec_, Oct 07 2022

%t With[{m = 20}, Range[0, m]! * CoefficientList[Series[Cosh[(Exp[4*x] - 1)/4], {x, 0, m}], x]] (* _Amiram Eldar_, Oct 07 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh((exp(4*x)-1)/4)))

%o (PARI) a(n) = sum(k=0, n\2, 4^(n-2*k)*stirling(n, 2*k, 2));

%Y Cf. A009153, A024430, A357649.

%Y Cf. A357617.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 07 2022