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A107635 McKay-Thompson series of class 32a for the Monster group.

%I A107635 #28 Feb 16 2025 08:32:57
%S A107635 1,3,3,4,9,12,15,21,30,43,54,69,94,123,153,193,252,318,391,486,609,
%T A107635 754,918,1119,1376,1680,2019,2432,2946,3540,4220,5034,6015,7157,8463,
%U A107635 9999,11835,13956,16374,19206,22542,26376,30750,35829,41745,48526,56250
%N A107635 McKay-Thompson series of class 32a for the Monster group.
%C A107635 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A107635 G. C. Greubel, <a href="/A107635/b107635.txt">Table of n, a(n) for n = 0..1000</a>
%H A107635 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 A107635 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A107635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A107635 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A107635 Expansion of q^(1/8) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^3 in powers of q.
%F A107635 Expansion of chi(x)^3 = phi(x) / psi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%F A107635 Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 - u) - 9*u*v.
%F A107635 Euler transform of period 4 sequence [3, -3, 3, 0, ...].
%F A107635 G.f.: Product_{k>0} (1 + (-x)^k)^-3.
%F A107635 a(n) = (-1)^n * A022598(n).
%F A107635 a(n) ~ exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A107635 G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018
%e A107635 G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 15*x^6 + 21*x^7 + ...
%e A107635 T32a = 1/q + 3*q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 15*q^47 + ...
%t A107635 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^2 / (QPochhammer[ x] QPochhammer[ x^4]))^3, {x, 0, n}]; (* _Michael Somos_, Jun 29 2014 *)
%t A107635 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%o A107635 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^3, n))};
%Y A107635 Cf. A022598.
%K A107635 nonn
%O A107635 0,2
%A A107635 _Michael Somos_, May 18 2005