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A303790 G.f. satisfies: 120*(1-216*x)*A(x) + (1-3*(1-216*x)^2)*A'(x) - (1-216*x)*(2-216*x)*x*A''(x) = 0, a(0)=1.

%I A303790 #39 Feb 08 2021 07:11:25
%S A303790 1,60,7380,1090320,176978340,30471320880,5461962826320,
%T A303790 1007754602437440,189974650649174820,36407481107391279600,
%U A303790 7068262344580438681680,1386636913539840633652800,274365765112318301005693200,54676607910763730416065374400
%N A303790 G.f. satisfies: 120*(1-216*x)*A(x) + (1-3*(1-216*x)^2)*A'(x) - (1-216*x)*(2-216*x)*x*A''(x) = 0, a(0)=1.
%C A303790 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).
%H A303790 G. C. Greubel, <a href="/A303790/b303790.txt">Table of n, a(n) for n = 0..250</a>
%H A303790 M. Kreshchuk and T. Gulden, <a href="https://arxiv.org/abs/1803.07566">The Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method</a>, arXiv:1803.07566 [hep-th], 2018.
%H A303790 Bradley Klee, <a href="/A303790/a303790.pdf">Proof Certificate</a>
%H A303790 Brad Klee, <a href="http://demonstrations.wolfram.com/DerivingHypergeometricPicardFuchsEquations/">Deriving Hypergeometric Picard-Fuchs Equations</a>, Wolfram Demonstrations Project (2018).
%F A303790 G.f.: 2F1(1/6, 5/6; 1; 432*x - 46656*x^2).
%F A303790 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.
%F A303790 a(n) ~ 6^(3*n) / (Pi*n). - _Vaclav Kotesovec_, May 01 2018
%e A303790 G.f. = 1 + 60*x + 7380*x^2 + 1090320*x^3 + 176978340*x^4 + 30471320880*x^5 + ... _Michael Somos_, Jun 22 2018
%t A303790 a[0] = 1; a[1] = 60;
%t A303790 a[n0_] := a[n0] = ReplaceAll[Dot[Divide[
%t A303790 {5-27*n+27*n^2,(5-3*n)*(-1+3*n)},18*n^2],
%t A303790 {216*a[n0-1],(216^2)*a[n0-2]}],n->n0]
%t A303790 a /@ Range[0, 15]
%t A303790 (* Second program: *)
%t A303790 CoefficientList[Series[Hypergeometric2F1[1/6, 5/6, 1, 432*x - 46656*x^2],{x,0,20}], x]
%Y A303790 Real Period: A113424.
%K A303790 nonn
%O A303790 0,2
%A A303790 _Bradley Klee_, Apr 30 2018