|
%I #20 Oct 04 2019 11:31:48
%S 1,-24,4372,96256,1240002,10698752,74428120,431529984,2206741887,
%T 10117578752,42616961892,166564106240,611800208702,2125795885056,
%U 7040425608760,22327393665024,68134255043715,200740384538624
%N Expansion of 64(g_n^(24) + g_n^(-24)) where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.
%D 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.
%H Seiichi Manyama, <a href="/A106207/b106207.txt">Table of n, a(n) for n = -1..10000</a>
%F a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017
%F 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
%e G.g. = 1/q - 24 + 4372q + 96256q^2 + 1240002q^3 + ...
%t 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 *)
%o (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))};
%o (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
%Y Cf. A007241 is unsigned version.
%Y Cf. A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.
%K sign
%O -1,2
%A _Michael Somos_, Apr 25 2005
|
