A381285 Expansion of e.g.f. 1/(1 - sin(2*x) / 2).

1, 1, 2, 2, -8, -104, -688, -3088, -128, 209536, 3145472, 29795072, 139389952, -1715047424, -60056147968, -1004215072768, -10305404960768, -1945682345984, 2949643589844992, 84438462955323392, 1458284922371571712, 12032890515685113856, -245515800089314459648

Offset: 0

Author

Seiichi Manyama, Feb 18 2025

Keywords

• sign

Crossrefs

Cf. A000111, A381286.

Cf. A136630.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[1/(1-Sin[2x]/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 09 2025 *)

PARI

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

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * binomial(n,2*k+1) * a(n-2*k-1).

a(n) = Sum_{k=0..n} k! * (2*i)^(n-k) * A136630(n,k), where i is the imaginary unit.