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%I #22 Jan 25 2023 10:04:38
%S 1,2,3,6,7,10,16,20,29,40,52,70,91,116,149,190,242,306,383,478,590,
%T 730,897,1096,1342,1630,1975,2390,2873,3448,4133,4932,5880,6994,8290,
%U 9814,11587,13650,16058,18848,22089,25842,30178,35186,40950,47594,55231,63996,74068,85592,98776,113864
%N Coefficients of replicable function number "32b".
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A058632/b058632.txt">Table of n, a(n) for n = -1..1000</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of q^(1/4)*(eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2 in powers of q. -_G. C. Greubel_, Jun 23 2018
%F a(n) ~ exp(sqrt(n/2)*Pi) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018
%F From _Michael Somos_, Jan 17 2023: (Start)
%F Expansion of (psi(x)/psi(-x^2))^2 = (phi(-x^4)/psi(-x))^2 = (chi(x)*chi(x^2))^2 = (chi(-x^4)/chi(-x))^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = f(t) where q = exp(2 Pi i t). (End)
%e T32b = 1/q + 2*q^3 + 3*q^7 + 6*q^11 + 7*q^15 + 10*q^19 + 16*q^23 + 20*q^27 + ...
%t eta[q_] := q^(1/24)*QPochhammer[q]; a := CoefficientList[Series[q^(1/4)*(eta[q^2]*eta[q^4]/(eta[q]*eta[q^8]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* _G. C. Greubel_, Jun 23 2018 *)
%t From _Michael Somos_, Jan 17 2023: (Start)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, q^(1/2)]/EllipticTheta[2, Pi/4, q])^2 / (2/q^(1/4)), {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, -q^4]/EllipticTheta[2, Pi/4, q^(1/2)])^2 *(2*q^(1/4)), {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ (QPochhammer[-q, q^2]*QPochhammer[-q^2, q^4])^2, {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ (QPochhammer[+q^4, q^8]/QPochhammer[+q, q^2])^2, {q, 0, n}]; (End)
%o (PARI) q='q+O('q^50); Vec((eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2) \\ _G. C. Greubel_, Jun 23 2018
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K nonn
%O -1,2
%A _N. J. A. Sloane_, Nov 27 2000
%E Terms a(6) onward added by _G. C. Greubel_, Jun 23 2018
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