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A058539 McKay-Thompson series of class 18d for the Monster group.

%I A058539 #33 Feb 16 2025 08:32:43
%S A058539 1,4,10,20,35,60,100,164,261,400,600,884,1291,1864,2656,3740,5205,
%T A058539 7184,9842,13388,18082,24244,32300,42784,56378,73928,96466,125284,
%U A058539 161981,208568,267524,341880,435343,552424,698666,880848,1107229,1387804,1734624,2162248
%N A058539 McKay-Thompson series of class 18d for the Monster group.
%C A058539 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A058539 G. C. Greubel, <a href="/A058539/b058539.txt">Table of n, a(n) for n = 0..1000</a>
%H A058539 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 A058539 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A058539 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A058539 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A058539 Expansion of (chi(-x^3) / chi(-x))^4 in powers of x where chi() is a Ramanujan theta function.
%F A058539 Expansion of q^(1/3) * c(q) * b(q^2) / (b(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
%F A058539 Expansion of q^(1/3) * (eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)))^4 in powers of q.
%F A058539 Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 8 * (u * v)^2 - (1 + u * v) * (u^2 - v) * (v^2 - u).
%F A058539 Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 9 * (u * v)^2 - (u - v^2 + u^2*v) * (v - u^2 + u*v^2).
%F A058539 Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 8 * u * v * w - (u^2 - v) * (w^2 - v).
%F A058539 Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u*v * (1 + 25 * u*v + u^2*v^2)^2 - (u^3 + v^3 + 10 * u*v * (1 + u*v))^2.
%F A058539 G.f. is a period 1 Fourier series which satisfies f(-1 / (54 t)) = f(t) where q = exp(2 Pi i t).
%F A058539 Convolution square of A103262. Convolution fourth power of A003105.
%F A058539 a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e A058539 1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 60*x^5 + 100*x^6 + 164*x^7 + ...
%e A058539 T18d = 1/q + 4*q^2 + 10*q^5 + 20*q^8 + 35*q^11 + 60*q^14 + 100*q^17 + ...
%t A058539 nmax = 50; CoefficientList[Series[Product[((1+x^(3*k-1))*(1+x^(3*k-2)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%t A058539 QP = QPochhammer; s = (QP[q^2]*(QP[q^3]/(QP[q]*QP[q^6])))^4 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o A058539 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^4, n))} /* _Michael Somos_, Mar 04 2012 */
%Y A058539 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058539 Cf. A003105, A103262.
%K A058539 nonn
%O A058539 0,2
%A A058539 _N. J. A. Sloane_, Nov 27 2000