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

Author

Peter Luschny, Apr 14 2014

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Programs

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}]

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

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)]

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)^*sqrt(Pi*(2*N+1))); here = 1 if n is even, 0 otherwise and 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