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.

0, 1, 40, 1876, 95072, 5045474, 276107408, 15444602248, 878268335296, 50588345910799, 2944021398570264, 172780225616034252, 10211876493716693664, 607169816926036666486, 36286222314596227018672, 2178246170438379512947864, 131270483744089714062036032

Offset

0,3

Comments

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]]}.

References

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

Links

Table of n, a(n) for n=0..16.

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.

Mathematica

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]];
qSer[nMax_] := Expand[Times[x, Normal[ Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/4, 3/4, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[20] /. {x -> k*x}, x] /. {k -> 64}

Crossrefs

Cf. A005797, A308835, A308837.

Sequence in context: A147520 A190926 A143314 * A189503 A190076 A278729

Adjacent sequences: A308833 A308834 A308835 * A308837 A308838 A308839

Keyword

nonn

Author

Bradley Klee, Jun 27 2019

Status

approved