A106183 Expansion of 1/sqrt(1-4*x-4*x^2+16*x^3).

1, 2, 8, 24, 88, 304, 1120, 4096, 15328, 57536, 218112, 830208, 3176704, 12196352, 46982144, 181452800, 702465536, 2724948992, 10589474816, 41217216512, 160657903616, 627019489280, 2449986043904, 9583049572352, 37519931654144

Offset: 0

Author

Paul Barry, Apr 24 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A067804. In general, a(n) = Sum_{k=0..floor(n/2)} C(2*k,k) * C(2*(n-2*k),n-2*k) * r^k has g.f. 1/sqrt(1-4*x-4*r*x^2+16*r*x^3).

Crossrefs

Cf. A026569.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-4*x-4*x^2+16*x^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

PARI

a(n) = sum(k=0, n\2, 2^(n-k)*binomial(2*k, k)*binomial(n-k, k)); \\ Seiichi Manyama, May 02 2025

Formula

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

D-finite with recurrence: n*a(n) +2*(1-2*n)*a(n-1) +4*(1-n)*a(n-2) +8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

a(n) ~ 2^(2*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Feb 03 2014

a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(2*k,k) * binomial(n-k,k). - Seiichi Manyama, May 02 2025