A098484 Expansion of 1/sqrt((1-x)^2-12x^4).

1, 1, 1, 1, 7, 19, 37, 61, 145, 397, 979, 2107, 4591, 10915, 26857, 63649, 146347, 339751, 808885, 1936717, 4588705, 10803133, 25559287, 60893551, 145231309, 345462145, 821110051, 1955736379, 4668132067, 11146642903, 26605635949

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. A098481, A098482, A098483.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^2-12*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

Formula

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

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 12*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

a(n) ~ sqrt(3) * (1+sqrt(1+8*sqrt(3)))^n / (sqrt(49+10*sqrt(3)-sqrt(397+884*sqrt(3))) * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Jun 23 2014