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A101127 McKay-Thompson series of class 12D for the Monster group.

%I A101127 #22 Feb 16 2025 08:32:55
%S A101127 1,8,28,64,134,288,568,1024,1809,3152,5316,8704,13990,22208,34696,
%T A101127 53248,80724,121240,180068,264448,384940,556064,796760,1132544,
%U A101127 1598789,2243056,3127360,4333568,5971922,8188096,11170160,15163392,20491033
%N A101127 McKay-Thompson series of class 12D for the Monster group.
%C A101127 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A101127 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A101127 Seiichi Manyama, <a href="/A101127/b101127.txt">Table of n, a(n) for n = 0..1000</a>
%H A101127 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%H A101127 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A101127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A101127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%F A101127 Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q.
%F A101127 Euler transform of period 4 sequence [8, -8, 8, 0, ...].
%F A101127 Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16.
%F A101127 G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8.
%F A101127 A007259(n) = (-1)^n * a(n). Convolution square of A112160.
%F A101127 a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A101127 Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Sep 12 2017
%F A101127 G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018
%e A101127 T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ...
%t A101127 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%t A101127 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* _Michael Somos_, Sep 12 2017 *)
%o A101127 (PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))};
%o A101127 (PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))};
%Y A101127 cf. A007259, A112160.
%K A101127 nonn
%O A101127 0,2
%A A101127 _Michael Somos_, Dec 02 2004