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%I #17 Aug 06 2017 13:11:20
%S 1,2,38,452,6470,92252,1352540,20056584,300546630,4537543340,
%T 68923356788,1052049129144,16123803193628,247959261273752,
%U 3824345320438520,59132290704871952,916312070771835462,14226520736453485260,221256270142955957252,3446310324328958045400,53753604366737011495220
%N a(n) = Sum_{k=0..n} binomial(2*n,2*k)^2.
%H Seiichi Manyama, <a href="/A098772/b098772.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = (binomial(4*n, 2*n)+(-1)^n*binomial(2*n, n))/2.
%F Recurrence: n*(n-1)*(2*n-1)*(5*n^2-15*n+11)*a(n)-4*(n-1)*(30*n^4-120*n^3+161*n^2-82*n+12)*a(n-1)-4*(4*n-7)*(2*n-3)*(4*n-5)*(5*n^2-5*n+1)*a(n-2) = 0.
%F a(n) ~ 2^(4*n-3/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 02 2017
%t Table[Sum[Binomial[2n,2k]^2,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jan 21 2016 *)
%o (Maxima) makelist((binomial(4*n,2*n)+(-1)^n*binomial(2*n,n))/2,n,0,12); /* _Emanuele Munarini_, Feb 01 2017 */
%o (PARI) a(n) = sum(k=0, n, binomial(2*n, 2*k)^2); \\ _Michel Marcus_, Feb 01 2017
%Y Cf. A037964, A000984.
%K nonn
%O 0,2
%A _Vladeta Jovovic_, Oct 03 2004