A390232 a(n) = (2*n)! * [x^(2*n)] (f(x)^2 + f(-x)^2)/2 and f(x) = 1/(cos(x) + sqrt(3) * sin(x)).

1, 20, 1456, 223040, 58575616, 23502402560, 13373331582976, 10243813494702080, 10163266698756161536, 12678333853101857177600, 19422906944247900899639296, 35848212199606304514894725120, 78455159096957357408585311584256, 200891413690245126967691882151280640

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

a(n) = (2*n)! * [x^(2*n)] (1 + 2*sin(x)^2)/(1 - 4*sin(x)^2)^2.

Let A(n,k) = (2*n)! * [x^(2*n)] (f(x)^k + f(-x)^k)/2. A(0,k) = 1 and A(0,k) = 1 and A(n,k) = 4*k*(k+1) * A(n-1,k+2) - k^2 * A(n-1,k) for n > 0. a(n) = A(n,2).

a(n) ~ 2^(4*n+2) * 3^(2*n+2) * n^(2*n + 3/2) / (exp(2*n) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Apr 17 2026

Mathematica

nmax = 20; CoefficientList[Series[(1 + 2*Sin[Sqrt[x]]^2)/(1 - 4*Sin[Sqrt[x]]^2)^2, {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Apr 17 2026 *)

Prog

(PARI) a(n) = my(x='x+O('x^(2*n+1))); (2*n)!*polcoef((1+2*sin(x)^2)/(1-4*sin(x)^2)^2, 2*n);

Crossrefs

Column k=2 of A395233.

Sequence in context: A127847 A363512 A193309 * A177601 A200900 A177603

Adjacent sequences: A390229 A390230 A390231 * A390233 A390234 A390235

Keyword

nonn

Author

Seiichi Manyama, Apr 17 2026

Status

approved