|
%I #23 Mar 31 2023 06:13:58
%S 1,2,3,4,5,10,15,22,29,36,53,72,99,128,160,212,272,354,448,556,703,
%T 874,1096,1356,1662,2050,2501,3060,3716,4492,5444,6550,7882,9436,
%U 11262,13460,16013,19034,22536,26616,31450,37048,43602,51164,59905
%N McKay-Thompson series of class 30D for the Monster group with a(0) = 2.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A205962/b205962.txt">Table of n, a(n) for n = -1..1000</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) * ((chi(-q^3) * chi(-q^15)) / (chi(-q) * chi(-q^5)))^2 in powers of q where chi() is a Ramanujan theta function.
%F Expansion of (eta(q^2) * eta(q^3) * eta(q^10) * eta(q^15) / (eta(q) * eta(q^5) * eta(q^6) * eta(q^30)))^2 in powers of q.
%F Euler transform of period 30 sequence [ 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - 4*u*v * (v - 1)^2. - _Michael Somos_, Jun 09 2012
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: (1/x) * (Product_{k>0} (1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))))^2.
%F a(n) = A058615(n) unless n=0. Convolution square of A058729.
%F a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015
%e 1/q + 2 + 3*q + 4*q^2 + 5*q^3 + 10*q^4 + 15*q^5 + 22*q^6 + 29*q^7 + ...
%t nmax = 50; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 06 2015 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^2] *eta[q^3]*eta[q^10]*eta[q^15]/(eta[q]*eta[q^5]*eta[q^6]*eta[q^30]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 06 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^30 + A)))^2, n))}
%Y Cf. A058729.
%Y Essentially the same as A058615.
%K nonn
%O -1,2
%A _Michael Somos_, Feb 02 2012
|
