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%I #30 Jan 02 2018 12:14:30
%S 1,2,2,3,4,6,9,12,16,21,28,36,47,60,76,96,120,150,185,228,280,342,416,
%T 504,608,732,878,1050,1252,1488,1765,2088,2464,2901,3408,3996,4676,
%U 5460,6364,7404,8600,9972,11545,13344,15400,17748,20424,23472,26938
%N McKay-Thompson series of class 36D for the Monster group with a(0) = 2.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A186964/b186964.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>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (1/q) * psi(q) * psi(q^9) / (psi(-q) * chi(q^3) * psi(q^18)) in power of q where psi(), chi() are Ramanujan theta functions.
%F Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^18)^3 / (eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^9) * eta(q^36)^2) in powers of q.
%F Euler transform of period 36 sequence [ 2, -1, 1, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 1, 0, 2, -2, 2, 0, 1, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 1, -1, 2, 0, ...].
%F G.f. is a period 1 Fourier series that satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A058647(n) = A187020(n) unless n=0. [typo corrected by _Vaclav Kotesovec_, Oct 10 2015]
%F a(n) ~ exp(2*sqrt(n)*Pi/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 10 2015
%e 1/q + 2 + 2*q + 3*q^2 + 4*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 16*q^7 + 21*q^8 + ...
%t nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^k) * (1+x^(6*k)) * (1+x^(9*k)) / ((1-x^(4*k)) * (1+x^(3*k)) * (1+x^(18*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 10 2015 *)
%t a[n_]:= SeriesCoefficient[(-q)^(1/8)*EllipticTheta[2, 0, Sqrt[q]]* EllipticTheta[2, 0, Sqrt[q^9]]/(QPochhammer[-q^3, q^6]*EllipticTheta[2, 0, I*Sqrt[q]]*EllipticTheta[2, 0, Sqrt[q^18]]), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* _G. C. Greubel_, Jan 02 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^3 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A)^2), n))}
%Y Cf. A058647, A187020.
%K nonn
%O -1,2
%A _Michael Somos_, Mar 06 2011
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