Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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