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 #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