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%I #18 Jul 19 2019 22:14:28
%S 0,1,15,279,5729,124554,2810718,65114402,1538182398,36887880105,
%T 895303119303,21943398532563,542209373589501,13489931811324550,
%U 337599511395854298,8491805574767197650,214548940430198454054,5441921826542937659088,138512110164878076019560
%N The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/27)^n follows from the series solutions of 2*T-d/dx(9*(1-x)*x*dT/dx)=0.
%C Also appears in Ramanujan's theory of elliptic functions, signature 3 (cf. A006480). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.
%D B. C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.
%H G. Almkvist et al., <a href="https://www.math.ru.nl/~wzudilin/PS/clausen-PEMS.pdf">Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations</a>, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275. [The article is on pages 273-295.]
%t G[nMax_]:=Dot[RecurrenceTable[{Dot[{(3*n-5)^2 (3*n-4)^2 (9*n-4), -18(n - 1)(40 - 197*n + 351*n^2 - 279*n^3 + 81*n^4),81(n - 1)*n^3*(9*n - 13)}, a[n-#] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/9}, a, {n, 0, nMax}], x^Range[0, nMax]];
%t qSer[nMax_] := Expand[Times[x, Normal[Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/3, 2/3, 1, x]]], {x, 0, nMax}]]]];
%t CoefficientList[(1/k)*qSer[20]/.{x->k*x},x]/.{k->27}
%Y Cf. A005797, A308836, A308837.
%K nonn
%O 0,3
%A _Bradley Klee_, Jun 27 2019
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