A187130 - OEIS

%I #17 Mar 12 2021 22:24:46

%S 1,-3,2,0,1,0,0,0,-2,0,-2,0,2,0,4,0,3,0,-4,0,-8,0,-4,0,5,0,14,0,7,0,

%T -8,0,-20,0,-12,0,14,0,28,0,17,0,-20,0,-44,0,-24,0,28,0,66,0,36,0,-40,

%U 0,-90,0,-52,0,56,0,124,0,71,0,-80,0,-176,0,-96,0,109

%N McKay-Thompson series of class 12I 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="/A187130/b187130.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 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).

%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)) / (psi(-q^3) * psi(q^6)) in powers of q where phi(), psi() are Ramanujan theta functions.

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

%F Euler transform of period 12 sequence [ -3, -1, -2, -2, -3, -2, -3, -2, -2, -1, -3, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187100.

%F Convolution inverse of A187100.

%e G.f. = 1/q - 3 + 2*q + q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...

%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, Pi/4, q^(1/2)] / (EllipticTheta[ 2, Pi/4, q^(3/2)] EllipticTheta[ 2, 0, q^3]), {q, 0, n}] // Simplify; (* _Michael Somos_, Apr 24 2015 *)

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

%Y Cf. A058487, A187100.

%K sign

%O -1,2

%A _Michael Somos_, Mar 05 2011