A308835 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.

0, 1, 15, 279, 5729, 124554, 2810718, 65114402, 1538182398, 36887880105, 895303119303, 21943398532563, 542209373589501, 13489931811324550, 337599511395854298, 8491805574767197650, 214548940430198454054, 5441921826542937659088, 138512110164878076019560

Offset: 0

Author

Bradley Klee, Jun 27 2019

Keywords

• nonn

Comments

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]]}.

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. [The article is on pages 273-295.]

Crossrefs

Cf. A005797, A308836, A308837.

Programs

Mathematica

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]];
qSer[nMax_] := Expand[Times[x, Normal[Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/3, 2/3, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[20]/.{x->k*x},x]/.{k->27}