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%I #17 Mar 12 2021 22:24:46
%S 1,2,0,0,2,0,0,0,-1,0,0,0,-2,0,0,0,3,0,0,0,2,0,0,0,-4,0,0,0,-4,0,0,0,
%T 5,0,0,0,8,0,0,0,-8,0,0,0,-10,0,0,0,11,0,0,0,12,0,0,0,-15,0,0,0,-18,0,
%U 0,0,22,0,0,0,26,0,0,0,-29,0,0,0,-34,0,0,0
%N McKay-Thompson series of class 16B 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="/A208603/b208603.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) * phi(q) / psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
%F Expansion of eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)^2 * eta(q^16)^2) in powers of q.
%F Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, ...].
%F G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (v - 2) * (u^2 - 4*u + 8).
%F G.f.: 2 + (1/q) * Product_{k>0} ((1 + q^(8*k - 4)) / (1 + q^(8*k)))^2.
%F a(4*n - 1) = A029839(n). a(4*n) = 0 unless n=0. a(4*n + 1) = a(4*n + 2) = 0. Convolution inverse of A208605.
%F a(n) = -(-1)^n * A185338(n).
%e G.f. = 1/q + 2 + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + 5*q^31 + ...
%t QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4]^2*QP[q^16]^2)) + O[q]^80; 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^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2), n))}
%Y Cf. A029839, A185338, A208605.
%K sign
%O -1,2
%A _Michael Somos_, Feb 29 2012
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