A106207 Expansion of 64(g_n^(24) + g_n^(-24)) where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant.

1, -24, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624

Offset

-1,2

References

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.

S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Formula

a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017

Expansion of (1 + (64*A)^2)/A, where A = (eta(q^2)/eta(q))^24, in powers of q. - G. C. Greubel, Jun 19 2018

Example

G.g. = 1/q - 24 + 4372q + 96256q^2 + 1240002q^3 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; A:= (eta[q^2]/eta[q])^24; a := CoefficientList[Series[q*(1 + (64*A)^2)/A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 19 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = prod(k=1, (n+1)\2, 1-x^(2*k-1), 1+x*O(x^n))^24; polcoeff( A + x^2*4096/A, n))};
(PARI) q='q+O('q^50); A = q*(eta(q^2)/eta(q))^24; Vec((1+(64*A)^2)/A) \\ G. C. Greubel, Jun 19 2018

Crossrefs

Cf. A007241 is unsigned version.

Cf. A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.

Sequence in context: A159399 A184687 A007241 * A100089 A151598 A003787

Adjacent sequences: A106204 A106205 A106206 * A106208 A106209 A106210

Keyword

sign

Author

Michael Somos, Apr 25 2005

Status

approved