cp's OEIS Frontend

A137635 a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.

%I A137635 #26 Jul 11 2024 14:38:01
%S A137635 1,2,10,46,226,1136,5810,30080,157162,826992,4376408,23267332,
%T A137635 124179570,664919780,3570265000,19216805476,103652442922,560127574340,
%U A137635 3031887311256,16435458039076,89213101943000,484839755040768,2637805800869740,14365506336197816
%N A137635 a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.
%C A137635 Diagonal of rational function 1/(1 - (x + y + x^2*y + x*y^2)). - _Gheorghe Coserea_, Aug 31 2018
%H A137635 Seiichi Manyama, <a href="/A137635/b137635.txt">Table of n, a(n) for n = 0..1000</a>
%F A137635 G.f.: A(x) = 1/sqrt(1 - 4x(1+x)^2).
%F A137635 D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +8*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Jan 14 2020
%F A137635 a(n) = binomial(2*n, n)*hypergeom([(1-2*n)/3, 2*(1-n)/3, -2*n/3], [1/2-n, 1/2-n], -3^3/2^4). - _Stefano Spezia_, Jul 11 2024
%t A137635 CoefficientList[Series[1/Sqrt[1 - 4*x*(1 + x)^2],{x,0,50}],x] (* _Stefano Spezia_, Sep 01 2018 *)
%t A137635 Table[Sum[Binomial[2k,k]Binomial[2k,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Dec 31 2018 *)
%t A137635 a[n_]:=Binomial[2n, n]HypergeometricPFQ[{(1-2*n)/3, 2(1-n)/3, -2n/3}, {1/2-n, 1/2-n}, -3^3/2^4]; Array[a,24,0] (* _Stefano Spezia_, Jul 11 2024 *)
%o A137635 (PARI) a(n)=sum(k=0,n,binomial(2*k,k)*binomial(2*k,n-k));
%o A137635 (PARI) a(n)=polcoeff(1/sqrt(1-4*x*(1+x +x*O(x^n))^2),n,x);  /* Using the g.f.: */
%Y A137635 Cf. A137634, A137636, A137637, A137638, A073157.
%K A137635 nonn
%O A137635 0,2
%A A137635 _Paul D. Hanna_, Jan 31 2008