%I #20 Mar 12 2021 22:24:44
%S 1,-1,0,-1,0,0,1,0,0,1,0,0,-1,0,0,0,0,0,1,0,0,-2,0,0,0,0,0,2,0,0,-3,0,
%T 0,1,0,0,4,0,0,-4,0,0,1,0,0,4,0,0,-6,0,0,1,0,0,5,0,0,-8,0,0,1,0,0,8,0,
%U 0,-10,0,0,2,0,0,11,0,0,-14,0,0,4,0,0,14,0,0,-19,0,0,4,0,0,17,0,0,-24,0,0,4,0,0,23
%N McKay-Thompson series of class 36B 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="/A132976/b132976.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 eta(q) * eta(q^4) * eta(q^18) / (eta(q^2) * eta(q^9) * eta(q^36)) in powers of q.
%F Expansion of psi(-q) / (q * psi(-q^9)) = -1 + chi(q^9)^3 / (q * chi(q^3)) in powers of q where psi(), chi() are Ramanujan theta functions.
%F Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u3 * u6 - (u1 + u2 + u1*u2) * (u3 + u6 + 3).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132975.
%F a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062244(n). a(2*n) = -A139032(n). a(6*n - 1) = A132179(n). a(6*n + 2) = -A092848(n).
%F a(n) = -(-1)^n * A143840(n). Convolution inverse of A132975.
%e G.f. = 1/q - 1 - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...
%t a[ n_] := If[ n < -1, 0, SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(9/2)], {q, 0, n}]];
%t QP = QPochhammer; s = QP[q] * QP[q^4] * (QP[q^18] / (QP[q^2] * QP[q^9] * QP[q^36])) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%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^18 + A) / (eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};
%Y Cf. A062244, A092848, A132975, A139032, A143840.
%K sign
%O -1,22
%A _Michael Somos_, Sep 07 2007