A162480 - OEIS

%I #16 Jul 20 2021 07:02:44

%S 1,4,21,126,797,5190,34439,231556,1572135,10754148,74001735,511686726,

%T 3552251429,24743806370,172853699427,1210514603212,8495774193707,

%U 59739915525288,420785972800187,2968344133842182,20967995689677183

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

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

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

%F 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

%F 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

%t CoefficientList[Series[1/((1-x)^2 Sqrt[1-4 x/(1-x)^4]),{x,0,20}],x] (* _Harvey P. Dale_, Jun 28 2017 *)

%t Table[Sum[Binomial[n+3k+1,4k+1]Binomial[2k,k],{k,0,n}],{n,0,100}] (* _Emanuele Munarini_, Aug 31 2017 *)

%o (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 */

%Y Partial sums of A162479.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 04 2009