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

%I #11 Jul 11 2020 02:33:35

%S 1,0,-4,32,-240,1792,-11840,26112,1589504,-57548800,1556757504,

%T -39250780160,973563695104,-24122607992832,596246557736960,

%U -14477682566889472,332039052050104320,-6425352382711857152,53086817854485692416,4505005802471597015040,-419037805969718712991744

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

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

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

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

%p gf := exp(1 - 1/(2*x + 1))/(2*x + 1): ser := series(gf, x, 32):

%p seq(n!*coeff(ser, x, n), n=0..20);

%p # Alternative:

%p a := proc(n) option remember; if n < 2 then 1 - n else

%p 4*(1 - n)*((n - 1)*a(n - 2) + a(n - 1)) fi end: seq(a(n), n=0..20);

%Y Cf. A331333, A009940.

%K sign

%O 0,3

%A _Peter Luschny_, Jan 19 2020