A113179 Expansion of 1/sqrt((1-2x)^2-8x^3).

1, 2, 4, 12, 40, 128, 408, 1328, 4384, 14560, 48576, 162816, 547936, 1850048, 6263680, 21257856, 72298240, 246345728, 840775424, 2873802240, 9835840512, 33704557568, 115622041600, 397032488960, 1364610270720, 4694145256448

Offset: 0

Author

Paul Barry, Oct 16 2005

Keywords

• easy
• nonn

Comments

In general, 1/sqrt((1-a*x)^2-4*b*x^3) expands to sum{k=0..floor(n/2), C(n-k,k)C(n-2k,k)b^k*a^(n-3k)}.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1837
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Crossrefs

Cf. A098479.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-2x)^2-8x^3],{x,0,30}],x] (* Harvey P. Dale, Dec 23 2017 *)

Formula

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

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +4*(n-1)*a(n-2) +4*(-2*n+3)*a(n-3)=0. [Belbachir]