A331334 a(n) = n! * [x^n] exp(1 - 1/(2*x + 1))/(2*x + 1).

1, 0, -4, 32, -240, 1792, -11840, 26112, 1589504, -57548800, 1556757504, -39250780160, 973563695104, -24122607992832, 596246557736960, -14477682566889472, 332039052050104320, -6425352382711857152, 53086817854485692416, 4505005802471597015040, -419037805969718712991744

Offset: 0

Author

Peter Luschny, Jan 19 2020

Keywords

• sign

Crossrefs

Cf. A331333, A009940.

Programs

Maple

gf := exp(1 - 1/(2*x + 1))/(2*x + 1): ser := series(gf, x, 32):
seq(n!*coeff(ser, x, n), n=0..20);
# Alternative:
a := proc(n) option remember; if n < 2 then 1 - n else
4*(1 - n)*((n - 1)*a(n - 2) + a(n - 1)) fi end: seq(a(n), n=0..20);

Formula

a(n) = 4*(1 - n)*((n - 1)*a(n - 2) + a(n - 1)).

a(n) = (-2)^n*Sum_{k=0..n} A331333(n, k)/(-2)^k.

a(n)/(-2)^n = n!*LaguerreL(n, 1) = A009940(n).