This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A037976 #16 Aug 07 2024 09:19:28
%S A037976 0,4,16,436,3200,78004,675808,15919320,150266880,3450748180,
%T A037976 34461586016,774842070600,8061900244736,178065876017176,
%U A037976 1912172640160960,41596867935469936,458156035085377536
%N A037976 a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2).
%D A037976 The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (See (3.75) on page 31.)
%H A037976 G. C. Greubel, <a href="/A037976/b037976.txt">Table of n, a(n) for n = 0..825</a>
%F A037976 From _G. C. Greubel_, Jun 22 2022: (Start)
%F A037976 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2.
%F A037976 a(n) = (1/4)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n)*(n+1)^2*A000108(n)^2 ).
%F A037976 G.f.: (1/4)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(2*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End)
%p A037976 A037976 := proc(n)
%p A037976 binomial(4*n,2*n)/4-(-1)^n*binomial(2*n,n)/4+(1-(-1)^n)*binomial(2*n,n)^2/4 ;
%p A037976 end proc:
%p A037976 seq(A037976(n),n=0..30) ; # _R. J. Mathar_, Jul 26 2015
%t A037976 With[{B=Binomial}, Table[(1/4)*(B[4*n,2*n] +B[2*n,n]^2 -2*(-1)^n*B[B[2*n,n] +1, 2]), {n,0,30}]] (* _G. C. Greubel_, Jun 22 2022 *)
%o A037976 (Magma) [(1/4)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n) +(1-(-1)^n)*(n+1)^2*Catalan(n)^2): n in [0..30]]; // _G. C. Greubel_, Jun 22 2022
%o A037976 (SageMath) b=binomial; [(1/4)*(b(4*n, 2*n) -(-1)^n*b(2*n, n) +(1-(-1)^n)*b(2*n, n)^2) for n in (0..30)] # _G. C. Greubel_, Jun 22 2022
%Y A037976 Cf. A000108, A037964, A037972.
%K A037976 nonn,easy
%O A037976 0,2
%A A037976 _N. J. A. Sloane_