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A082304 McKay-Thompson series of class 16d for the Monster group.

%I A082304 #28 Feb 16 2025 08:32:48
%S A082304 1,-2,-1,2,3,-2,-4,4,5,-8,-8,10,11,-12,-15,18,22,-26,-29,34,38,-42,
%T A082304 -51,56,66,-78,-85,98,109,-120,-139,156,176,-202,-222,250,279,-306,
%U A082304 -346,384,429,-482,-530,590,650,-714,-797,876,972,-1080,-1180,1304,1431,-1562,-1728,1892,2078,-2290,-2496
%N A082304 McKay-Thompson series of class 16d for the Monster group.
%C A082304 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A082304 Seiichi Manyama, <a href="/A082304/b082304.txt">Table of n, a(n) for n = 0..1000</a>
%H A082304 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 A082304 J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275. see page 273.
%H A082304 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A082304 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A082304 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A082304 Expansion of phi(-q) / psi(q^2) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A082304 Expansion of q^(1/4) * (eta(q) / eta(q^4))^2 in powers of q.
%F A082304 Euler transform of period 4 sequence [ -2, -2, -2, 0, ...].
%F A082304 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A001936. - _Michael Somos_, Jul 04 2014
%F A082304 Given g.f. A(x), then B(q) = A(q)^4 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u * (16 + u) * (16 + v). - _Michael Somos_, Jul 04 2014
%F A082304 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (4 + u*v)^2. - _Michael Somos_, Aug 13 2007
%F A082304 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u*v * (16 + u^2*v^2)^2 - (u+v)^2 * (u^2 - 6*u*v + v^2)^2. - _Michael Somos_, Jul 04 2014
%F A082304 G.f.: Product_{k>0} ((1 - x^k) / (1 - x^(4*k)))^2.
%F A082304 a(n) = (-1)^n * A029839(n). Convolution inverse of A001936. - _Michael Somos_, Jul 04 2014
%F A082304 abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Feb 07 2023
%e A082304 T16d = 1/q - 2*q^3 - q^7 + 2*q^11 + 3*q^15 - 2*q^19 - 4*q^23 + 4*q^27 + ...
%t A082304 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^4])^2, {x, 0, n}]; (* _Michael Somos_, Jul 04 2014 *)
%o A082304 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^4 + A))^2, n))};
%Y A082304 Cf. A001936, A029839.
%K A082304 sign
%O A082304 0,2
%A A082304 _Michael Somos_, Apr 08 2003