A331612 E.g.f.: exp(1 / (2 - sec(x)) - 1) (even powers only).

1, 1, 14, 481, 30449, 3064306, 448104029, 89621046061, 23468873468054, 7786478152466221, 3190021872763911149, 1580829351026679822586, 931656913226081002622489, 643808850722810399312420281, 515431991397502094847830786174, 473171296200788822261644150349881

Offset

0,3

Links

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * A002114(k) * a(n-k).

a(n) ~ 2^(2*n) * 3^(2*n + 1/8) * exp(-5/12 + sqrt(3)/(4*Pi) + 2*3^(1/4)*sqrt(n/Pi) - 2*n) * n^(2*n - 1/4) / Pi^(2*n + 1/4). - Vaclav Kotesovec, Jan 26 2020

Mathematica

nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Sec[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
e[0] = 1; e[n_] := e[n] = (-1)^n (1 - Sum[(-1)^j Binomial[2 n, 2 j] 3^(2 (n - j)) e[j], {j, 0, n - 1}]); A002114[n_] := e[n]/2^(2 n + 1); a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A002114[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
With[{nn=40}, Take[CoefficientList[Series[Exp[1/(2-Sec[x])-1], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Aug 08 2023 *)

Crossrefs

Cf. A000364, A002114, A217502, A331611.

Sequence in context: A208563 A200061 A171208 * A275092 A275348 A317635

Adjacent sequences: A331609 A331610 A331611 * A331613 A331614 A331615

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 22 2020

Status

approved