A162480 Expansion of 1/((1-x)^2*sqrt(1-4x/(1-x)^4)).

1, 4, 21, 126, 797, 5190, 34439, 231556, 1572135, 10754148, 74001735, 511686726, 3552251429, 24743806370, 172853699427, 1210514603212, 8495774193707, 59739915525288, 420785972800187, 2968344133842182, 20967995689677183

Offset: 0

Author

Paul Barry, Jul 04 2009

Keywords

• easy
• nonn

Crossrefs

Partial sums of A162479.

Programs

Mathematica

CoefficientList[Series[1/((1-x)^2 Sqrt[1-4 x/(1-x)^4]),{x,0,20}],x] (* Harvey P. Dale, Jun 28 2017 *)
Table[Sum[Binomial[n+3k+1,4k+1]Binomial[2k,k],{k,0,n}],{n,0,100}] (* Emanuele Munarini, Aug 31 2017 *)

Maxima

makelist(sum(binomial(n+3*k+1,4*k+1)*binomial(2*k,k),k,0,n),n,0,12); /* Emanuele Munarini, Aug 31 2017 */

Formula

G.f.: 1/((1-x)^2-2x/((1-x)^2-x/((1-x)^2-x/((1-x)^2-... (continued fraction);

a(n) = Sum_{k=0..n} C(n+3k+1,n-k)*A000984(k).

D-finite with recurrence: n*a(n) +4*(1-2n)*a(n-1) +6*(n-1)*a(n-2) +2*(3-2n)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

G.f.: 1/sqrt(1-8*t+6*t^2-4*t^3+t^4). Remark: using this form of the g.f., it is easy to prove the above recurrence. - Emanuele Munarini, Aug 31 2017