A098481 Expansion of 1/sqrt((1-x)^2 - 12*x^3).

1, 1, 1, 7, 19, 37, 115, 361, 937, 2599, 7777, 22195, 62701, 182647, 531829, 1534903, 4461571, 13034917, 38015899, 110994193, 325011151, 952442557, 2792471239, 8198275933, 24093817531, 70852613041, 208516575043, 614145137137

Offset: 0

Author

Paul Barry, Sep 10 2004

Keywords

• easy
• nonn

Comments

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

Links

• G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)

Crossrefs

Cf. A098479, A098480.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^2-12x^3],{x,0,40}],x] (* Harvey P. Dale, Jun 02 2011 *)

PARI

Vec(1/sqrt((1-x)^2 - 12*x^3) + O(x^50)) \\ G. C. Greubel, Jan 30 2017

Formula

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

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 6*(2*n-3)*a(n-3). - Vaclav Kotesovec, Jun 23 2014

a(n) ~ 3^(n+1) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 23 2014