A143840 - OEIS

%I #17 Mar 12 2021 22:24:45

%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,

%U 0,0,-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 18D 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="/A143840/b143840.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 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 Expansion of eta(q^2)^2 * eta(q^9) / (eta(q) * eta(q^18)^2) in powers of q.

%F Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128770.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (u^2 + 3) - (u + v)^2.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 3*u + 3) * (v^2 - 3*v + 3) - v^3.

%F a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062242(n). a(2*n) = A139032(n). a(6*n + 2) = A092848(n). a(6*n - 1) = A132179(n).

%F A132976(n) = -(-1)^n * a(n). Convolution inverse of A124243.

%F G.f.: 1 + x^(-1) * Product_{k>0} (1 - x^(18*k - 9))^3 / (1 - x^(6*k - 3)).

%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, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(9/2)], {q, 0, n}]];

%t QP = QPochhammer; s = QP[q^2]^2*(QP[q^9]/(QP[q]*QP[q^18]^2)) + 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^2 + A)^2 * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)^2), n))};

%Y Cf. A062242, A092848, A128770, A124243, A132179, A132976, A139032.

%K sign

%O -1,22

%A _Michael Somos_, Sep 02 2008