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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. 3
0, 1, 312, 107604, 39073568, 14645965026, 5609733423408, 2182717163349896, 859521859502348352, 341679883727799750159, 136868519056531319862408, 55173969942211048781835468, 22360181278518828446785034976, 9103073677708423854325869548662 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..13.

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[{(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}

CROSSREFS

Cf. A005797, A308835, A308836.

Sequence in context: A156406 A283492 A112729 * A239818 A290181 A282792

Adjacent sequences:  A308834 A308835 A308836 * A308838 A308839 A308840

KEYWORD

nonn

AUTHOR

Bradley Klee, Jun 27 2019

STATUS

approved

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Last modified July 9 22:46 EDT 2020. Contains 335570 sequences. (Running on oeis4.)