A187145 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

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (phi(q) * psi(q)) / (psi(q^3) * psi(q^6)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2)^7 * eta(q^3) / (eta(q)^3 * eta(q^4)^2 * eta(q^6) * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -4, 3, -2, 2, -4, 3, 0, ...].

a(2*n) = 0 unless n=0. a(2*n - 1) = A058487(n).

a(n) = -(-1)^n * A187130(n). Convolution inverse is A228447. - Michael Somos, Sep 05 2015

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[ 3, 0, q] EllipticTheta[ 2, 0, q^(1/2)]/(EllipticTheta[ 2, 0, q^(3/2)] EllipticTheta[ 2, 0, q^3]), {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A058487, A187130, A228447.

Sequence in context: A133209 A144553 A187130 * A285700 A290693 A131290

Adjacent sequences: A187142 A187143 A187144 * A187146 A187147 A187148

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved