A184990 - OEIS

%I #28 Mar 12 2021 22:24:46

%S 1,1,0,2,-1,-2,4,-2,-2,6,-4,-4,10,-6,-8,16,-9,-10,24,-14,-16,36,-20,

%T -24,53,-30,-32,76,-43,-48,108,-60,-68,150,-84,-92,206,-114,-128,280,

%U -155,-172,376,-208,-228,504,-276,-304,668,-366,-400,878,-480,-524,1148

%N McKay-Thompson series of class 24C for the Monster group with a(0) = 1.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A184990/b184990.txt">Table of n, a(n) for n = -1..5000</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 (1/q) * psi(q) * phi(-q^4) / (psi(-q^3) * psi(-q^6)) in powers of q where phi(), psi() are Ramanujan theta functions.

%F Expansion of eta(q^2)^2 * eta(q^4)^2 / (eta(q) * eta(q^3) * eta(q^8)* eta(q^24)) in powers of q.

%F Euler transform of period 24 sequence [ 1, -1, 2, -3, 1, 0, 1, -2, 2, -1, 1, -2, 1, -1, 2, -2, 1, 0, 1, -3, 2, -1, 1, 0, ...].

%F a(n) = A058573(n) unless n = 0.

%e 1/q + 1 + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 + ...

%t QP = QPochhammer; s=QP[q^2]^2*(QP[q^4]^2/(QP[q]*QP[q^3]*QP[q^8]*QP[q^24]))+ O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^8 + A)* eta(x^24 + A)), n))}

%Y Cf. A058573.

%K sign

%O -1,4

%A _Michael Somos_, Feb 05 2012