A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)*exp(x) + 3*exp(-x) - 4)/ (4x^2).

1, 6, 8, 60, 180, 1680, 8064, 90720, 604800, 7983360, 68428800, 1037836800, 10897286400, 186810624000, 2324754432000, 44460928512000, 640237370572800, 13516122267648000, 221172909834240000, 5109094217170944000, 93666727314800640000, 2350183339898634240000, 47726800133326110720000

Offset: 0

Author

Pierre-Alain Sallard, Dec 14 2018

Keywords

• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A060593 (even bisection, shifted), A028242 (denominator minus 1), A030451 (denominator, shifted), A107991 (Expansion of a similar function), A073743.

Programs

GAP

List([0..25],n->(4*Factorial(n+2))/(2*n+5+3*(-1)^n)); # Muniru A Asiru, Dec 20 2018

Maple

a:=n->factorial(n+2)/(3*floor(n/2)-n+2): seq(a(n),n=0..25); # Muniru A Asiru, Dec 20 2018

Mathematica

Table[4*Factorial[n + 2]/(2*n + 5 + 3*(-1)^n), {n, 0, 25}]
(* or *)
Function[x, 1/x] /@
CoefficientList[Series[(Exp[x]/4 + 3/4*Exp[-x] + x/2*Exp[x] - 1)/x^2, {x, 0, 20}], x]

PARI

a(n)={(4*(n+2)!)/(5 + 3*(-1)^n + 2*n)} \\ Andrew Howroyd, Dec 14 2018

PARI

my(x='x + O('x^30)); Vec(apply(x->1/x, ((1+2*x)*exp(x) + 3*exp(-x) - 4)/ (4*x^2))) \\ Michel Marcus, Dec 19 2018

Formula

a(n) = (n+2)!/(3*floor(n/2)-n+2).

a(n) = (4*(n+2)!)/(2n+5+3*(-1)^n).

a(n) = 4/([x^n]((exp(x)*(1+2x)+3*exp(-x)-4)/x^2)).

a(n) = (n+2)!/(A028242(n)+1).

a(n) = (n+2)!/A030451(n+1).

a(n) ~ sqrt(Pi/2)/72*exp(-n)*n^(n-1/2)*(1705 - 264*n + 288*n^2). - Stefano Spezia, Aug 11 2025

Sum_{n>=0} 1/a(n) = 3*cosh(1)/2 - 1. - Amiram Eldar, Aug 15 2025