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%I #10 Jul 29 2018 20:46:21
%S 1,1,3,27,441,11529,442827,23444883,1636819569,145703137041,
%T 16106380394643,2164638920874507,347592265948756521,
%U 65724760945840254489,14454276753061349098587,3658147171522531111996803,1055646229815910768764248289,344553616791279239828059918881
%N a(n) = coefficient of x^(2*n)/(2*n)! in exp( integral ( sn(x, 1/2) / cd(x, 1/2) ) dx).
%H G. C. Greubel, <a href="/A286306/b286306.txt">Table of n, a(n) for n = 0..249</a>
%F Given e.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n) / (2*n)!, then 0 = 1 + 2*A'^2 - A*A''.
%F Given e.g.f. A(x), then A'(x) / A(x) = B(x) where B() is the e.g.f. for A242240.
%F Given e.g.f. A(x), 1 / A(x) = A(-x).
%F A159600(n) = (-1)^n * a(n). A159601(n) = -(-1)^n * a(n) if n>0.
%F A190904(2*n) = A193541(n) = (-1)^floor(n/2) * a(n). A193544(n) = (-1)^floor((n+1)/2) * a(n).
%e G.f. = 1 + x + 3*x^2 + 27*x^3 + 441*x^4 + 11529*x^5 + 442827*x^6 + ...
%e E.g.f. = 1 + 1*x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! + ...
%t a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Exp[ Integrate[ JacobiSN[x, 1/2] / JacobiCD[x, 1/2], x]], {x, 0, m}]]];
%t a:= With[{nmax = 110}, CoefficientList[Series[Exp[Integrate[JacobiSN[x, 1/2]/JacobiCD[x, 1/2], x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jul 29 2018 *)
%o (PARI) {a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( exp( intformal( serreverse( intformal( (1 + x^4 + x * O(x^m))^(-1/2))))), m))};
%Y Cf. A159600, A159601, A190904, A193541, A193544, A242240.
%K nonn
%O 0,3
%A _Michael Somos_, May 05 2017
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