A098483 Expansion of 1/sqrt((1-x)^2-8x^4).

1, 1, 1, 1, 5, 13, 25, 41, 85, 205, 473, 985, 2021, 4365, 9785, 21673, 46965, 101581, 222745, 492665, 1087237, 2388749, 5251065, 11587529, 25633045, 56697933, 125345113, 277283353, 614212133, 1361824525, 3020426681, 6700678377

Offset: 0

Author

Paul Barry, Sep 10 2004

Keywords

• easy
• nonn

Comments

1/sqrt((1-x)^2-4rx^4) expands to sum{k=0..floor(n/2), binomial(n-2k,k)binomial(n-3k,k)r^k}

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A098480, A098482, A098484.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^2-8*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

PARI

a(n) = sum(k=0, n\2, binomial(n-2*k, k)*binomial(n-3*k, k)*2^k) \\ Michel Marcus, Jul 24 2013

Formula

a(n)=sum{k=0..floor(n/2), binomial(n-2k, k)binomial(n-3k, k)2^k}.

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 8*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

a(n) ~ (1+sqrt(1+8*sqrt(2)))^n / (sqrt(33+10*sqrt(2)-sqrt(265+596*sqrt(2))) * sqrt(Pi*n) * 2^(n-3/2)). - Vaclav Kotesovec, Jun 23 2014