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A286306 a(n) = coefficient of x^(2*n)/(2*n)! in exp( integral ( sn(x, 1/2) / cd(x, 1/2) ) dx). 1
1, 1, 3, 27, 441, 11529, 442827, 23444883, 1636819569, 145703137041, 16106380394643, 2164638920874507, 347592265948756521, 65724760945840254489, 14454276753061349098587, 3658147171522531111996803, 1055646229815910768764248289, 344553616791279239828059918881 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..249

FORMULA

Given e.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n) / (2*n)!, then 0 = 1 + 2*A'^2 - A*A''.

Given e.g.f. A(x), then A'(x) / A(x) = B(x) where B() is the e.g.f. for A242240.

Given e.g.f. A(x), 1 / A(x) = A(-x).

A159600(n) = (-1)^n * a(n). A159601(n) = -(-1)^n * a(n) if n>0.

A190904(2*n) = A193541(n) = (-1)^floor(n/2) * a(n). A193544(n) = (-1)^floor((n+1)/2) * a(n).

EXAMPLE

G.f. = 1 + x + 3*x^2 + 27*x^3 + 441*x^4 + 11529*x^5 + 442827*x^6 + ...

E.g.f. = 1 + 1*x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! + ...

MATHEMATICA

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}]]];

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 *)

PROG

(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))};

CROSSREFS

Cf. A159600, A159601, A190904, A193541, A193544, A242240.

Sequence in context: A159601 A193541 A193544 * A285239 A111844 A277352

Adjacent sequences:  A286303 A286304 A286305 * A286307 A286308 A286309

KEYWORD

nonn

AUTHOR

Michael Somos, May 05 2017

STATUS

approved

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Last modified October 20 19:57 EDT 2019. Contains 328269 sequences. (Running on oeis4.)