%I #9 Jun 22 2022 20:44:24

%S 1,1,53,262,8885,60626,1778966,14168988,383358645,3355615450,

%T 85990654178,803232328548,19780031677718,193873026294052,

%U 4629016098160220,47101568276955512,1096960888092571317,11503661742608944170,262435310495071434602

%N a(n) = (binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2)/2.

%D Identity (3.71) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 30.

%H G. C. Greubel, <a href="/A036911/b036911.txt">Table of n, a(n) for n = 0..825</a>

%F a(n) = (1/2)*(binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2).

%F From _G. C. Greubel_, Jun 22 2022: (Start)

%F a(n) = Sum_{k=0..n} binomial(2*n, 2*k)^2.

%F a(n) = (1/2)*((2*n+1)*A000108(2*n) + (-1)^n*(n+1)^2*A000108(n)^2).

%F G.f.: (1/2)*( sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) + (2/Pi)*EllipticK(-16*x]) ). (End)

%t Table[(Binomial[4n,2n]+(-1)^n Binomial[2n,n]^2)/2,{n,0,20}] (* _Harvey P. Dale_, May 22 2013 *)

%o (Magma) [(1/2)*((2*n+1)*Catalan(2*n) + (-1)^n*(n+1)^2*Catalan(n)^2): n in [0..30]]; // _G. C. Greubel_, Jun 22 2022

%o (SageMath) b=binomial; [(1/2)*(b(4*n, 2*n) + (-1)^n*b(2*n, n)^2) for n in (0..30)] # _G. C. Greubel_, Jun 22 2022

%Y Cf. A000108, A037980.

%K nonn

%O 0,3

%A _N. J. A. Sloane_