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A308836 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/64)^n follows from the series solutions of 3*T-d/dx(16*(1-x)*x*dT/dx)=0.

%I A308836 #8 Jul 02 2019 18:42:32
%S A308836 0,1,40,1876,95072,5045474,276107408,15444602248,878268335296,
%T A308836 50588345910799,2944021398570264,172780225616034252,
%U A308836 10211876493716693664,607169816926036666486,36286222314596227018672,2178246170438379512947864,131270483744089714062036032
%N A308836 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/64)^n follows from the series solutions of 3*T-d/dx(16*(1-x)*x*dT/dx)=0.
%C A308836 Also appears in Ramanujan's theory of elliptic functions, signature 4 (cf. A000897). 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 A308836 B.C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.
%H A308836 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.
%t A308836 G[nMax_] := Dot[RecurrenceTable[ {Dot[{(4*n - 7)^2 (4*n - 5)^2 (8*n - 3), -16 (n - 1) (105 - 562*n + 1056*n^2 - 864*n^3 + 256*n^4), 256 (n - 1) n^3 (8*n - 11)},  a[n - #] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/8}, a, {n, 0, nMax}], x^Range[0, nMax]];
%t A308836 qSer[nMax_] := Expand[Times[x, Normal[ Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/4, 3/4, 1, x]]], {x, 0, nMax}]]]];
%t A308836 CoefficientList[(1/k)*qSer[20] /. {x -> k*x}, x] /. {k -> 64}
%Y A308836 Cf. A005797, A308835, A308837.
%K A308836 nonn
%O A308836 0,3
%A A308836 _Bradley Klee_, Jun 27 2019