A187148 - OEIS

%I #29 Mar 12 2021 22:24:46

%S 1,-3,6,-4,-3,12,-8,-12,30,-20,-30,72,-46,-60,156,-96,-117,300,-188,

%T -228,552,-344,-420,1008,-603,-732,1770,-1048,-1245,2976,-1776,-2088,

%U 4908,-2900,-3420,7992,-4658,-5460,12756,-7408,-8583,19944,-11564,-13344,30756

%N McKay-Thompson series of class 12B 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="/A187148/b187148.txt">Table of n, a(n) for n = -1..1000</a>

%H 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, (1994), 5175-5193.

%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 (1/q) * (chi(q^3)^3 / chi(q))^3 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q) * eta(q^4) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3))^3 in powers of q.

%F Convolution cube of A062244. a(3*n) = -3 * A164617(n). a(3*n + 1) = 6 * A132977(n).

%F G.f.: (Product_{k>0} (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)))^3.

%e G.f. = 1/q - 3 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 + ...

%t a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q^3, q^6]^3 / QPochhammer[ -q, q^2])^3, {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3))^3, n))};

%Y Cf. A187146, A187147, A062244, A112148, A132977, A164617.

%K sign

%O -1,2

%A _Michael Somos_, Mar 05 2011