OFFSET
0,2
COMMENTS
The surface "u = 2H = p^2 + q^2 - (4/27)*q^6" determines a Picard-Fuchs equation, "5*u*T(u) + 9*(3*u^2-1)*T'(u) + 9*(u^2-1)*u*T''(u) = 0", (cf. link to "Proof Certificate"). The Picard-Fuchs differential equation transforms to the defining relation by "u->1-216*x". G.f. A(x) generates coefficients of the complex period-energy function, while the real period-energy function can be written in terms of hypergeometric A113424. These results agree with Kreshchuk and Gulden, as "d/du(5*u*T(u) + 9*(3*u^2-1)*T'(u) + 9*(u^2-1)*u*T''(u)) = 5*T(u) + 59*u*T'(u) + 18*(3*u^2-1)*T''(u) + 9*u*(u^2-1)*T'''(u) = 0" (cf. Eq. 16).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
M. Kreshchuk and T. Gulden, The Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method, arXiv:1803.07566 [hep-th], 2018.
Bradley Klee, Proof Certificate
Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018).
FORMULA
G.f.: 2F1(1/6, 5/6; 1; 432*x - 46656*x^2).
D-finite with recurrence a(0) = 1; a(1) = 60; a(n) = (c1/c0)*216*a(n-1) + (c2/c0)*216^2*a(n-2); with c1 = 5-27*n+27*n^2; c2 = (5-3*n)*(-1+3*n); c0 = 18*n^2.
a(n) ~ 6^(3*n) / (Pi*n). - Vaclav Kotesovec, May 01 2018
EXAMPLE
G.f. = 1 + 60*x + 7380*x^2 + 1090320*x^3 + 176978340*x^4 + 30471320880*x^5 + ... Michael Somos, Jun 22 2018
MATHEMATICA
a[0] = 1; a[1] = 60;
a[n0_] := a[n0] = ReplaceAll[Dot[Divide[
{5-27*n+27*n^2, (5-3*n)*(-1+3*n)}, 18*n^2],
{216*a[n0-1], (216^2)*a[n0-2]}], n->n0]
a /@ Range[0, 15]
(* Second program: *)
CoefficientList[Series[Hypergeometric2F1[1/6, 5/6, 1, 432*x - 46656*x^2], {x, 0, 20}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Bradley Klee, Apr 30 2018
STATUS
approved