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%I #25 Mar 12 2021 22:24:42
%S 1,0,2,3,4,6,9,12,16,21,28,36,47,60,76,96,120,150,185,228,280,342,416,
%T 504,608,732,878,1050,1252,1488,1765,2088,2464,2901,3408,3996,4676,
%U 5460,6364,7404,8600,9972,11545,13344,15400,17748,20424,23472,26938,30876
%N McKay-Thompson series of class 36D for the Monster simple group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A058647/b058647.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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%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 q^(-1) * f(q^3) * phi(q^3) / (phi(-q^2) * psi(q^9)) in powers of q where f(), phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 13 2011
%F Expansion of eta(q^4) * eta(q^6)^8 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2) in powers of q. - _Michael Somos_, Feb 13 2011
%F Euler transform of period 36 sequence [ 0, 2, 3, 1, 0, -3, 0, 1, 2, 2, 0, -1, 0, 2, 3, 1, 0, -2, 0, 1, 3, 2, 0, -1, 0, 2, 2, 1, 0, -3, 0, 1, 3, 2, 0, 0, ...]. - _Michael Somos_, Feb 13 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Feb 13 2011
%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Dec 03 2015
%e T36D = 1/q + 2*q + 3*q^2 + 4*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 16*q^7 + ...
%t a[ n_] := SeriesCoefficient[ 2 q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 3, 0, q^3] / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t nmax = 60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(3*k))^2 * (1+x^(3*k))^5 / ((1-x^(2*k)) * (1+x^(6*k))^3 * (1-x^(9*k)) * (1+x^(9*k))^2), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Dec 03 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^8 * eta(x^9 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 * eta(x^18 + A)^2), n))} /* _Michael Somos_, Feb 13 2011 */
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K nonn
%O -1,3
%A _N. J. A. Sloane_, Nov 27 2000
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