%I #11 Jan 30 2020 21:29:18
%S 1,0,3,6,81,540,7155,85050,1346625,22339800,431331075,9004668750,
%T 208178118225,5199538043700,140664514065075,4080315642653250,
%U 126613733680058625,4180226398201854000,146399020309066399875,5419213146765629961750,211446723837565171580625
%N E.g.f.: 1 / ((1 + x) * sqrt(1 - 2*x)).
%F a(n) = n! * Sum_{k=0..n} (-1)^(n - k) * (2*k - 1)!! / k!.
%F D-finite with recurrence: a(n) +(-n+1)*a(n-1) -(2*n-1)*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jan 25 2020
%F a(n) ~ 2^(n + 3/2) * n^n / (3*exp(n)). - _Vaclav Kotesovec_, Jan 26 2020
%t nmax = 20; CoefficientList[Series[1/((1 + x) Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[n! Sum[(-1)^(n - k) (2 k - 1)!!/k!, {k, 0, n}], {n, 0, 20}]
%o (PARI) a(n) = {n! * sum(k=0, n, (-1)^(n - k) * (2*k)! / (2^k*k!^2))} \\ _Andrew Howroyd_, Jan 16 2020
%o (PARI) seq(n) = {Vec(serlaplace(1 / ((1 + x) * sqrt(1 - 2*x + O(x*x^n)))))} \\ _Andrew Howroyd_, Jan 16 2020
%Y Cf. A001147, A005359, A034430, A052585, A317618.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 16 2020
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