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%I #12 Aug 04 2022 05:10:56
%S 1,3,16,116,1038,10922,131256,1766592,26253702,426173906,7492270416,
%T 141661870088,2864168171596,61621248390756,1404853103594128,
%U 33815954626749600,856680253728250950,22777071459869216850,633968368216974945600,18430976777427663053400
%N a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k) * (n-k)!.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(x))^2.
%F a(n) ~ n^n * exp(4*sqrt(n) - n - 2) / sqrt(8*Pi) * (1 + 55/(24*sqrt(n))). - _Vaclav Kotesovec_, Aug 04 2022
%F Recurrence: n*a(n) = (3*n^2 + n - 1)*a(n-1) - (n-1)^2*(3*n + 1)*a(n-2) + (n-2)^2*(n-1)^2*a(n-3). - _Vaclav Kotesovec_, Aug 04 2022
%t Table[Sum[Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 19}]
%t Table[n! HypergeometricPFQ[{1/2, -n}, {1, 1}, -4], {n, 0, 19}]
%t nmax = 19; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^2 * binomial(2*k,k) * (n-k)!); \\ _Michel Marcus_, Jul 17 2020
%Y Cf. A000984, A002720, A002893.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 16 2020
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