This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331403 #11 Jan 30 2020 21:29:18
%S A331403 1,0,3,6,81,540,7155,85050,1346625,22339800,431331075,9004668750,
%T A331403 208178118225,5199538043700,140664514065075,4080315642653250,
%U A331403 126613733680058625,4180226398201854000,146399020309066399875,5419213146765629961750,211446723837565171580625
%N A331403 E.g.f.: 1 / ((1 + x) * sqrt(1 - 2*x)).
%F A331403 a(n) = n! * Sum_{k=0..n} (-1)^(n - k) * (2*k - 1)!! / k!.
%F A331403 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 A331403 a(n) ~ 2^(n + 3/2) * n^n / (3*exp(n)). - _Vaclav Kotesovec_, Jan 26 2020
%t A331403 nmax = 20; CoefficientList[Series[1/((1 + x) Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A331403 Table[n! Sum[(-1)^(n - k) (2 k - 1)!!/k!, {k, 0, n}], {n, 0, 20}]
%o A331403 (PARI) a(n) = {n! * sum(k=0, n, (-1)^(n - k) * (2*k)! / (2^k*k!^2))} \\ _Andrew Howroyd_, Jan 16 2020
%o A331403 (PARI) seq(n) = {Vec(serlaplace(1 / ((1 + x) * sqrt(1 - 2*x + O(x*x^n)))))} \\ _Andrew Howroyd_, Jan 16 2020
%Y A331403 Cf. A001147, A005359, A034430, A052585, A317618.
%K A331403 nonn
%O A331403 0,3
%A A331403 _Ilya Gutkovskiy_, Jan 16 2020