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%I #36 Mar 12 2021 22:24:46
%S 1,1,2,3,4,6,9,12,16,21,28,36,47,60,76,96,120,150,185,228,280,342,416,
%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) = 1.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Seiichi Manyama, <a href="/A187020/b187020.txt">Table of n, a(n) for n = -1..1000</a>
%H K. Bringmann and H. Swisher, <a href="http://dx.doi.org/10.1090/S0002-9939-07-08735-7">On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions</a>, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2317-2326. (MR230255) see p. 2325 Appendix 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, (1994) 5175-5193.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015
%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>
%F Expansion of (1/q) * chi(-q^9) * chi(-q^18) / (chi(-q) * chi(-q^2)) in powers of q where chi() is a Ramanujan theta function.
%F Expansion of eta(q^4) * eta(q^9) / (eta(q) * eta(q^36)) in powers of q.
%F Euler transform of period 36 sequence [ 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u + v) * (u + v + u*v) * (u^2 + u*v + v^2) - (u + u^2 + u^3) * (v + v^2 + v^3).
%F G.f. A(x) = ( G(x^36) * H(x) - x^7 * H(x^36) * G(x) ) / ( G(x^9) * H(x^4) - x * H(x^9) * G(x^4) ) where G(x) (g.f. of A003114) and H(x) (g.f. of A003106) are Rogers-Ramanujan functions [Bringmann + Swisher, (A.12)]
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A058647(n) = A186964(n) unless n=0. A128128(n) = a(2*n). A000005(n) = a(2*n - 1).
%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e 1/q + 1 + 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^(4*k)) * (1-x^(9*k)) / ((1-x^k) * (1-x^(36*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%t QP = QPochhammer; s = QP[q^4]*(QP[q^9]/(QP[q]*QP[q^36])) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^36 + A)), n))}
%Y Cf. A000005, A003106, A003114, A058647, A128128, A186964.
%K nonn
%O -1,3
%A _Michael Somos_, Mar 06 2011
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