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%I #20 Nov 29 2020 22:36:01
%S 1,5,53,761,12661,229705,4410665,88127485,1813270645,38158684745,
%T 817458330553,17767242718285,390819348043369,8683822363169933,
%U 194618212789162733,4394243766346694161,99862206804817230965,2282427331053360624713
%N a(n) = Sum_{k=0..n} binomial(2*n-2*k,n-k)^2 * binomial(n,k)^2.
%D Cf. M. Petkovsek et al., A=B, Peters, p. ix.
%H Reinhard Zumkeller, <a href="/A036916/b036916.txt">Table of n, a(n) for n = 0..500</a>
%F (n - 1)*(144*n^3 - 864*n^2 + 1693*n - 1075)*n^3*a(n) - 2*(n - 1)*(2592*n^6 - 19440*n^5 + 56322*n^4 - 80296*n^3 + 60004*n^2 - 23017*n + 3580)*a(n - 1) + (42336*n^7 - 423360*n^6 + 1769838*n^5 - 4006912*n^4 + 5293968*n^3 - 4062414*n^2 + 1661406*n - 274520)*a(n - 2) - 2*(34848*n^5 - 261360*n^4 + 741842*n^3 - 984642*n^2 + 598948*n - 127215)*(n - 2)^2*a(n - 3) + 225*(144*n^3 - 432*n^2 + 397*n - 102)*(n - 2)^2*(n - 3)^2*a(n - 4) = 0 - _Vladeta Jovovic_, Jul 15 2004
%F a(n) ~ 5^(2*n+2) / (16 * (Pi*n)^(3/2)). - _Vaclav Kotesovec_, Mar 02 2014
%t Table[Sum[Binomial[2n-2k,n-k]^2 Binomial[n,k]^2,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Mar 31 2013 *)
%o (Haskell)
%o a036916 n = sum $ map
%o (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 n k)^2) [0..n]
%o -- _Reinhard Zumkeller_, May 24 2012
%Y Cf. A036915, A036917.
%Y Cf. A007318.
%Y Row n=4 of A275784.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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