A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

Mathematica

With[{nn=20}, CoefficientList[Series[1/Sqrt[1-2x Cos[2x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 07 2026 *)

Prog

(PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*(2*I)^(n-k)*a185951(n, k));

Crossrefs

Cf. A352252, A385284.

Cf. A001586, A235134, A380155, A385281.

Cf. A001147, A185951.

Sequence in context: A219210 A218863 A082394 * A308645 A086889 A373174

Adjacent sequences: A385280 A385281 A385282 * A385284 A385285 A385286

Keyword

sign,changed

Author

Seiichi Manyama, Jun 24 2025

Status

approved