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A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2*k)! / (2^k * (k!)^3 * (n - k)!).

%I A331817 #16 Sep 08 2022 08:46:25
%S A331817 1,2,9,66,681,9090,148905,2889810,64805265,1648535490,46896669225,
%T A331817 1475099460450,50831084252025,1904311245686850,77061447551313225,
%U A331817 3349828945512299250,155672917524626126625,7701743926471878533250,404153655359180645543625
%N A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2*k)! / (2^k * (k!)^3 * (n - k)!).
%H A331817 Robert Israel, <a href="/A331817/b331817.txt">Table of n, a(n) for n = 0..380</a>
%F A331817 E.g.f.: 1 / sqrt(1 - 4*x + 3*x^2).
%F A331817 a(n) = Sum_{k=0..n} binomial(n,k)^2 * (2*k - 1)!! * (n - k)!.
%F A331817 a(n) = n! * 2F1(1/2, -n; 1; -2).
%F A331817 a(n) ~ 3^(n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Jan 28 2020
%F A331817 D-finite with recurrence a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n). - _Robert Israel_, Feb 17 2020
%p A331817 f:= gfun:-rectoproc({a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n), a(0)=1, a(1)=2},a(n), remember):
%p A331817 map(f, [$0..30]); # _Robert Israel_, Feb 17 2020
%t A331817 Table[n!^2 Sum[(2 k)!/(2^k k!^3 (n - k)!), {k, 0, n}], {n, 0, 18}]
%t A331817 nmax = 18; CoefficientList[Series[1/Sqrt[1 - 4 x + 3 x^2], {x, 0, nmax}], x] Range[0, nmax]!
%t A331817 Table[n! Hypergeometric2F1[1/2, -n, 1, -2], {n, 0, 18}]
%o A331817 (PARI) seq(n) = {Vec(serlaplace(1/(sqrt(1 - 4*x + 3*x^2 + O(x*x^n)))))} \\ _Andrew Howroyd_, Jan 27 2020
%o A331817 (Magma) [(Factorial(n))^2*&+[Factorial(2*k)/(2^k*(Factorial(k))^3*Factorial(n-k)):k in [0..n]]:n in [0..18]]; // _Marius A. Burtea_, Jan 27 2020
%Y A331817 Cf. A000165, A001147, A032031, A034430, A052577.
%K A331817 nonn
%O A331817 0,2
%A A331817 _Ilya Gutkovskiy_, Jan 27 2020