A395275 a(n) = (2*n)! * [x^(2*n)] cos(x) * cos(sqrt(2)*x) / cos(2*x).

1, 1, 25, 1225, 111313, 16242481, 3475772713, 1025518574137, 399001996871713, 197932331128444129, 121932807678038896825, 91323550356887465693737, 81722586234545124861515761, 86114383343209950647083860241, 105540155267496333911548805876425, 148852872072708090199843045982344345

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = (-1)^n * A001541(n) - Sum_{k=0..n-1} (-4)^(n-k) * binomial(2*n,2*k) * a(k).

a(n) ~ cos(Pi/2^(3/2)) * 2^(6*n + 5/2) * n^(2*n + 1/2) / (sqrt(Pi) * exp(2*n) * Pi^(2*n)). - Vaclav Kotesovec, Apr 18 2026

Mathematica

nmax = 20; CoefficientList[Series[Cos[Sqrt[x]] * Cos[Sqrt[2*x]] / Cos[2*Sqrt[x]], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Apr 18 2026 *)

Prog

(PARI) a001541(n) = polchebyshev(n, 1, 3);
a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i*a001541(i)-sum(j=0, i-1, (-4)^(i-j)*binomial(2*i, 2*j)*v[j+1])); v;

Crossrefs

Cf. A001541, A395276.

Sequence in context: A012809 A014769 A012851 * A181719 A096330 A174751

Adjacent sequences: A395272 A395273 A395274 * A395276 A395277 A395278

Keyword

nonn

Author

Seiichi Manyama, Apr 18 2026

Status

approved