A331818 E.g.f.: exp(1 - sec(x)) (even powers only).

1, -1, -2, -1, 253, 12854, 668053, 39148199, 2456262898, 130790155859, -3853032641387, -4312625669814166, -1531200886955161127, -489884373969089299201, -159097972223555719000922, -54064488830901650420384521, -19284261543086608770504566147

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2020

Keywords

• sign

Crossrefs

Cf. A000364, A005046, A217502, A260884.

Programs

Mathematica

nmax = 16; Table[(CoefficientList[Series[Exp[1 - Sec[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[2 n - 1, 2 k - 1] Abs[EulerE[2 k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

PARI

a(n) = {polcoef(serlaplace(exp(1 - 1/cos(x + O(x^(2*n + 1))))), 2*n)} \\ Andrew Howroyd, Jan 27 2020

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(2*n-1,2*k-1) * A000364(k) * a(n-k).