A205826 McKay-Thompson series of class 30A for the Monster group with a(0) = -3.

1, -3, 3, -1, 0, 0, 0, -3, 9, -9, 3, -3, 9, -12, 15, -18, 12, -6, 18, -39, 48, -46, 36, -24, 37, -75, 96, -90, 81, -78, 99, -165, 222, -199, 147, -150, 208, -306, 411, -424, 345, -327, 442, -606, 735, -756, 645, -606, 837, -1182, 1386, -1405, 1281, -1188, 1451

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

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * ((chi(-q) * chi(-q^15)) / (chi(-q^3) * chi(-q^5)))^3 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q) * eta(q^6) * eta(q^10) * eta(q^15) / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30)))^3 in powers of q.

Euler transform of period 30 sequence [ -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, -3, 0, -3, 0, 0, 0, -3, 0, -3, 0, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, ...].

Convolution cube of A112215. a(n) = A058612(n) unless n=0.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018

Example

1/q - 3 + 3*q - q^2 - 3*q^6 + 9*q^7 - 9*q^8 + 3*q^9 - 3*q^10 + 9*q^11 + ...

Mathematica

QP = QPochhammer; s = (QP[q]*QP[q^6]*QP[q^10]*(QP[q^15] / (QP[q^2]*QP[q^3]* QP[q^5]*QP[q^30])))^3 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)))^3, n))}

Crossrefs

Cf. A058612, A112215.

Sequence in context: A051343 A356326 A261477 * A131193 A350479 A130632

Adjacent sequences: A205823 A205824 A205825 * A205827 A205828 A205829

Keyword

sign

Author

Michael Somos, Feb 01 2012

Status

approved