A253282 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/6)), m = 5 - sqrt(24), and sn() is a Jacobi elliptic function.

1, 1, 2, 12, 124, 1844, 39288, 1134928, 42346256, 1985443536, 114380311072, 7938644848832, 653292526793664, 62901472582993984, 7005466255571255168, 893590563265303934208, 129425758313629525647616, 21124489015640181154724096, 3859303832272520341300756992

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

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

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.

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(1/6)), m = 5 - sqrt(24), and sn() is a Jacobi elliptic function.

Example

G.f. = 1 + x + 2*x^2 + 12*x^3 + 124*x^4 + 1844*x^5 + 39288*x^6 + ...
E.g.f. = x + x^3/6 + x^5/60 + x^7/420 + 31*x^9/90720 + 461*x^11/9979200 + ...

Mathematica

a[ n_] := If[ n < 0, 0, With[{t = Sqrt[-1/2 - Sqrt[1/6]], m = 5 - Sqrt[24]}, SeriesCoefficient[ JacobiSN[ t x, m] / t, {x, 0, 2 n + 1}] (2 n + 1)! // Simplify]];

Prog

(PARI) {a(n) = my(A, c); if( n<0, 0, A = x + x^3/6 + x^5/60; 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))};

Crossrefs

Cf. A144853, A253649.

Sequence in context: A227458 A035351 A209627 * A201470 A349268 A366821

Adjacent sequences: A253279 A253280 A253281 * A253283 A253284 A253285

Keyword

nonn

Author

Michael Somos, May 02 2015

Status

approved