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 A253649 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function. 1

%I

%S 1,1,0,-10,-80,0,17600,418000,0,-496672000,-23576960000,0,

%T 91442700800000,7255463564800000,0,-69994087116448000000,

%U -8354181454767104000000,0,169165728883243642880000000,28336045031124313753600000000,0,-1072156342430107319243161600000000

%N Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

%F The e.g.f. A(x) = y satisfies 0 = 2 - 2 * y'*y' + y*y'' + y^2.

%F The e.g.f. A(x) satisfies 0 = A(x) * A(y) * A(x-y) + A(y) * A(z) * A(y-z) - A(x) * A(z) * A(x-z) - A(x-y) * A(x-z) * A(y-z) for all x, y, z.

%F E.g.f.: Sum_{k>=0} a(k) * x^(2*k+1) / (2*k+1)! = sn(t * x, m) / t where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

%F a(3*n + 2) = 0. a(n) = (-1)^floor(n/3) * A063902(n) unless n == 2 (mod 3).

%e G.f. = 1 + x - 10*x^3 - 80*x^4 + 17600*x^6 + 418000*x^7 - 496672000*x^9 - ...

%e E.g.f. = x + x^3/6 - x^7/504 - x^9/4536 + x^13/353808 + 19/59439744*x^15 + ...

%t a[ n_] := If[ n < 0, 0, With[{t = Sqrt[-1/2 - Sqrt[1/3]], m = -7 + Sqrt[48]}, SeriesCoefficient[ JacobiSN[ t x, m] / t, {x, 0, 2 n + 1}] (2 n + 1)! // Simplify]];

%o (PARI) {a(n) = my(A, c); if( n<0, 0, A = x + x^3/6; for(k=3, n, A += O(x^(2*k+2)); A = x + intformal( intformal( 2*(A'^2 - 1) / A - A)); c = polcoeff( A, 2*k + 1) * k / (k-2); A = truncate( A + O(x^(2*k))) + c * x^(2*k+1)); (2*n + 1)! * polcoeff( A, 2*n + 1))};

%Y Cf. A063902, A144853, A253282.

%K sign

%O 0,4

%A _Michael Somos_, May 02 2015

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Last modified October 15 12:23 EDT 2019. Contains 328026 sequences. (Running on oeis4.)