A249512 Expansion of 1/(1-x*sqrt(4*x^2+1)-2*x^2).

1, 1, 3, 7, 15, 33, 75, 169, 375, 835, 1875, 4203, 9375, 20931, 46875, 104919, 234375, 523737, 1171875, 2621545, 5859375, 13098001, 29296875, 65523597, 146484375, 327500413, 732421875, 1637918089

Offset: 0

Author

Vladimir Kruchinin, Oct 31 2014

Keywords

• nonn

Programs

Maple

# Using function CompInv from A357588.
1, CompInv(27, n -> simplify(GegenbauerC(n-1, 1-n, 3/2))); # Peter Luschny, Oct 05 2022

Mathematica

CoefficientList[Series[1/(1-x*Sqrt[4*x^2+1]-2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 31 2014 *)

Maxima

a(n) := if n=0 then 1  else sum(k*4^(n-k)*binomial(n/2,n-k),k,1,n)/n;

Sage

def a(n):
    if is_odd(n):
        return simplify((4^(n-1)*binomial(n/2, n-1)*hypergeometric([2, 1-n], [2-n/2], -1/4))/n)
    return 3*5^(n//2-1) if n>0 else 1
[a(n) for n in (0..27)] # Peter Luschny, Oct 31 2014

Formula

a(n) = sum(k = 1..n, k*4^(n-k)*binomial(n/2,n-k))/n, a(0)=1.

a(n) ~ 3 * 5^(n/2-1). - Vaclav Kotesovec, Oct 31 2014

a(n) = 3 * 5^(n/2-1) if n is even and n>0 else a(n) = ((4^(n-1)* binomial(n/2, n-1)*hypergeometric([2, 1-n],[2-n/2], -1/4))/n). - Peter Luschny, Oct 31 2014

D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +(n+11)*a(n-2) +(n+10)*a(n-3) +20*(n-4)*a(n-4) +20*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 25 2020