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%I #29 Mar 12 2021 22:24:42
%S 1,2,2,4,7,10,14,20,27,36,50,64,84,110,140,180,229,288,360,452,560,
%T 692,854,1044,1275,1554,1884,2276,2745,3296,3950,4724,5630,6696,7946,
%U 9408,11115,13108,15422,18112,21238,24850,29034,33864,39429,45844,53224,61696
%N McKay-Thompson series of class 32B for the Monster group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A058630/b058630.txt">Table of n, a(n) for n = 0..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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of chi(x)^2 * chi(x^2) * chi(x^4)^2 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Oct 25 2013
%F Contribution from _Michael Somos_, Aug 08 2012 (Start)
%F Expansion of f(x) * f(x^4) / (f(-x) * f(-x^16)) = psi(x) * psi(x^4) / (psi(-x) * psi(x^8)) = chi(x) * chi(x^4) * chi(-x^8) / chi(-x) = (phi(x) * phi(x^4) / (phi(-x) * psi(x^8)))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of q^(1/2) * (eta(q^2) * eta(q^8))^3 / (eta(q) * eta(q^4) * eta(q^16))^2 in powers of q.
%F Euler transform of period 16 sequence [ 2, -1, 2, 1, 2, -1, 2, -2, 2, -1, 2, 1, 2, -1, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = f(t) where q = exp(2 Pi i t).
%F Convolution square is A214035. (end)
%F Expansion of psi(-x^2)^2 / (phi(-x) * psi(x^8)) = phi(x) * phi(x^4) / psi(-x^2)^2 in powers of x. - _Michael Somos_, Dec 14 2014
%F a(n) ~ exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 14*x^6 + 20*x^7 + 27*x^8 + ...
%e T32B = 1/q + 2*q + 2*q^3 + 4*q^5 + 7*q^7 + 10*q^9 + 14*q^11 + 20*q^13 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 QPochhammer[ -x^4, x^4] / (QPochhammer[-x^2, x^2] QPochhammer[-x^8, x^8]^2), {x, 0, n}]; (* _Michael Somos_, Oct 25 2013 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2 QPochhammer[ -x^2, x^4] QPochhammer[ -x^4, x^8]^2, {x, 0, n}]; (* _Michael Somos_, Dec 14 2014 *)
%t nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(8*k)))^3 / ((1-x^k) * (1-x^(4*k)) * (1-x^(16*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^8 + A))^3 / (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^2, n))}; /* _Michael Somos_, Aug 08 2012 */
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y Cf. A214035.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000
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