A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/432)^n follows from the series solutions of 5*T-d/dx(36*(1-x)*x*dT/dx)=0.

0, 1, 312, 107604, 39073568, 14645965026, 5609733423408, 2182717163349896, 859521859502348352, 341679883727799750159, 136868519056531319862408, 55173969942211048781835468, 22360181278518828446785034976, 9103073677708423854325869548662

Offset: 0

Author

Bradley Klee, Jun 27 2019

Keywords

• nonn

Comments

Also appears in Ramanujan's theory of elliptic functions, signature 6 (cf. A113424). 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]]}.

References

• B.C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

Links

• G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275.

Crossrefs

Cf. A005797, A308835, A308836.

Programs

Mathematica

G[nMax_] := Dot[RecurrenceTable[{Dot[{(6*n - 11)^2 (6*n - 7)^2 (18*n - 5), -36 (n - 1) (385 - 2426*n + 4968*n^2 - 4248*n^3 + 1296*n^4), 1296 (n - 1) n^3 (18*n - 23)},
a[n - #] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 13/18}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x,Normal[ Series[Exp[Divide[G[nMax], Hypergeometric2F1[1/6, 5/6, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[12] /. {x -> k*x}, x] /. {k -> 432}