A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

1, 1, 3, 14, 93, 796, 8343, 103424, 1479993, 24008656, 435364683, 8726775584, 191601310293, 4572794295616, 117871476051423, 3263515787807744, 96591500816346993, 3043368045293138176, 101702692426476460563, 3592948632452749243904, 133794496537591022166093

Offset: 0

Author

Seiichi Manyama, Jun 24 2025

Keywords

• nonn

Crossrefs

Cf. A000111, A385307.

Cf. A380015, A385304, A385308, A385310.

Cf. A001147, A001586, A007289, A136630.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2025 *)

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

a(n) ~ 2^(n+1) * 3^(n + 1/4) * n^n / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025