|
%I #15 Mar 12 2021 22:24:46
%S 1,-3,3,-1,0,0,0,-3,9,-9,3,-3,9,-12,15,-18,12,-6,18,-39,48,-46,36,-24,
%T 37,-75,96,-90,81,-78,99,-165,222,-199,147,-150,208,-306,411,-424,345,
%U -327,442,-606,735,-756,645,-606,837,-1182,1386,-1405,1281,-1188,1451
%N McKay-Thompson series of class 30A for the Monster group with a(0) = -3.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A205826/b205826.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) * chi(-q^15)) / (chi(-q^3) * chi(-q^5)))^3 in powers of q where chi() is a Ramanujan theta function.
%F Expansion of (eta(q) * eta(q^6) * eta(q^10) * eta(q^15) / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30)))^3 in powers of q.
%F Euler transform of period 30 sequence [ -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, -3, 0, -3, 0, 0, 0, -3, 0, -3, 0, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, ...].
%F Convolution cube of A112215. a(n) = A058612(n) unless n=0.
%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 29 2018
%e 1/q - 3 + 3*q - q^2 - 3*q^6 + 9*q^7 - 9*q^8 + 3*q^9 - 3*q^10 + 9*q^11 + ...
%t QP = QPochhammer; s = (QP[q]*QP[q^6]*QP[q^10]*(QP[q^15] / (QP[q^2]*QP[q^3]* QP[q^5]*QP[q^30])))^3 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)))^3, n))}
%Y Cf. A058612, A112215.
%K sign
%O -1,2
%A _Michael Somos_, Feb 01 2012
|
