A206298 McKay-Thompson series of class 24C for the Monster group with a(0) = -2.

1, -2, 0, 2, -1, -2, 4, -2, -2, 6, -4, -4, 10, -6, -8, 16, -9, -10, 24, -14, -16, 36, -20, -24, 53, -30, -32, 76, -43, -48, 108, -60, -68, 150, -84, -92, 206, -114, -128, 280, -155, -172, 376, -208, -228, 504, -276, -304, 668, -366, -400, 878, -480, -524, 1148

Offset

-1,2

Comments

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

Essentially the same as A206299, A184990 and A058573. - R. J. Mathar, Mar 11 2012

Links

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

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^4)) / (phi(-q^3) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q)^2 * eta(q^6) * eta(q^8)^2 * eta(q^12) / (eta(q^2) * eta(q^3)^2 * eta(q^4) * eta(q^24)^2) in powers of q.

Euler transform of period 24 sequence [ -2, -1, 0, 0, -2, 0, -2, -2, 0, -1, -2, 0, -2, -1, 0, -2, -2, 0, -2, 0, 0, -1, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3 / f(t) where q = exp(2 Pi i t).

a(n) = A058573(n) unless n = 0.

Example

1/q - 2 + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 + ...

Mathematica

QP = QPochhammer; s = QP[q]^2*QP[q^6]*QP[q^8]^2*(QP[q^12] / (QP[q^2]* QP[q^3]^2*QP[q^4]*QP[q^24]^2)) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A)^2), n))}

Crossrefs

Cf. A058573.

Sequence in context: A125922 A283306 A182893 * A076608 A068461 A130456

Adjacent sequences: A206295 A206296 A206297 * A206299 A206300 A206301

Keyword

sign

Author

Michael Somos, Feb 05 2012

Status

approved