A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).

1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).

For a recurrence see the Sage program.

a(n) = 2^n*A057977(n)

a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).

a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).

From Peter Luschny, Jan 31 2015: (Start)

a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).

a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).

a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).

a(n) ~ 2^(n+N)/((n+1)^<n>*sqrt(Pi*(2*N+1))); here <n> = 1 if n is even, 0 otherwise and N = n+<n>+1. (End)

Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

Maple

A240558 := n -> 2^n*n!/((iquo(n, 2)+1)*iquo(n, 2)!^2):
seq(A240558(n), n=0..30);

Mathematica

Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x, 0, n}], {n, 0, 22}]

Prog

(Sage)
def A240558():
    x, n = 1, 1
    while True:
        yield x
        m = 2*n if is_odd(n) else 8/(n+2)
        x *= m
        n += 1
a = A240558(); [next(a) for i in range(36)]
(PARI) x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017

Crossrefs

Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.

Sequence in context: A098656 A138611 A171459 * A163896 A350813 A262698

Adjacent sequences: A240555 A240556 A240557 * A240559 A240560 A240561

Keyword

nonn,easy

Author

Peter Luschny, Apr 14 2014

Status

approved