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A278069

a(n) = hypergeometric([n, -n], [], 1).

1, 0, 3, -32, 465, -8544, 190435, -4996032, 150869313, -5155334720, 196677847971, -8286870547680, 382200680031313, -19152276311294112, 1036167879649219395, -60195061159370501504, 3737352803142621672705, -246970483156591884266112, 17306865588065164490357443

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

a(-n) = a(n).

a(n) = n! [x^n] (2*x*exp(h(x)/2))/(4*x-h(x)) with h(x) = sqrt(4*x+1)-1.

a(n) ~ (-1)^n * 2^(2*n-1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Nov 10 2016

(-9-3*n)*a(n+1)+(12*n^2+53*n+52)*a(n+2)+(4*n^2+33*n+63)*a(n+3)+(n+4)*a(n+4) = 0. - Robert Israel, Nov 10 2016

a(n) = ((2*n-1)*a(n-2)-8*(1+n*(n-2))*a(n-1))/(2*n-3) for n>=2. - Peter Luschny, Nov 10 2016

a(n) = n! * [x^n] exp(x)/(1 + x)^n. - Ilya Gutkovskiy, Apr 07 2018

MAPLE

a := n -> hypergeom([n, -n], [], 1): seq(simplify(a(n)), n=0..18);

# Alternative:

a := proc(n) option remember; `if`(n<2, 1-n,

((2*n-1)*a(n-2)-8*(1+n*(n-2))*a(n-1))/(2*n-3)) end:

seq(a(n), n=0..18);

MATHEMATICA

Table[HypergeometricPFQ[{n, -n}, {}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 10 2016 *)

a={}; For[n=0, n<19, n++, AppendTo[a, (-1)^n*Sum[(-1)^(j-n+1-Mod[n, 2])*Product[(2*n-k)*k/(n-k+1), {k, j, n}], {j, 1, n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

PROG

(SageMath)

def a():

a, b, c, d, h, e = 1, 0, 1, 8, 8, 0

yield a

while True:

yield b

e = c; c += 2

a, b = b, (c*a - h*b)//e

d += 16; h += d

A278069 = a()

[next(A278069) for _ in range(19)]