A187130 McKay-Thompson series of class 12I for the Monster group with a(0) = -3.

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

Offset

-1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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.

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.

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

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.

Convolution inverse of A187100.

Example

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 + ...

Mathematica

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 *)

Prog

(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))};

Crossrefs

Cf. A058487, A187100.

Sequence in context: A059033 A133209 A144553 * A187145 A285700 A290693

Adjacent sequences: A187127 A187128 A187129 * A187131 A187132 A187133

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved