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 A385306 #17 Aug 09 2025 14:58:38
%S A385306 1,1,3,14,93,796,8343,103424,1479993,24008656,435364683,8726775584,
%T A385306 191601310293,4572794295616,117871476051423,3263515787807744,
%U A385306 96591500816346993,3043368045293138176,101702692426476460563,3592948632452749243904,133794496537591022166093
%N A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).
%F A385306 a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
%F A385306 a(n) ~ 2^(n+1) * 3^(n + 1/4) * n^n / (exp(n) * Pi^(n + 1/2)). - _Vaclav Kotesovec_, Jun 28 2025
%t A385306 With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 09 2025 *)
%o A385306 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A385306 a001147(n) = prod(k=0, n-1, 2*k+1);
%o A385306 a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));
%Y A385306 Cf. A000111, A385307.
%Y A385306 Cf. A380015, A385304, A385308, A385310.
%Y A385306 Cf. A001147, A001586, A007289, A136630.
%K A385306 nonn
%O A385306 0,3
%A A385306 _Seiichi Manyama_, Jun 24 2025