OFFSET
0,3
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
KEYWORD
nonn
AUTHOR
Michael Somos, May 02 2015
STATUS
approved