A208603 McKay-Thompson series of class 16B for the Monster group with a(0) = 2.

1, 2, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, -4, 0, 0, 0, 5, 0, 0, 0, 8, 0, 0, 0, -8, 0, 0, 0, -10, 0, 0, 0, 11, 0, 0, 0, 12, 0, 0, 0, -15, 0, 0, 0, -18, 0, 0, 0, 22, 0, 0, 0, 26, 0, 0, 0, -29, 0, 0, 0, -34, 0, 0, 0

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) * phi(q) / psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)^2 * eta(q^16)^2) in powers of q.

Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, ...].

G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (v - 2) * (u^2 - 4*u + 8).

G.f.: 2 + (1/q) * Product_{k>0} ((1 + q^(8*k - 4)) / (1 + q^(8*k)))^2.

a(4*n - 1) = A029839(n). a(4*n) = 0 unless n=0. a(4*n + 1) = a(4*n + 2) = 0. Convolution inverse of A208605.

a(n) = -(-1)^n * A185338(n).

Example

G.f. = 1/q + 2 + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + 5*q^31 + ...

Mathematica

QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4]^2*QP[q^16]^2)) + O[q]^80; 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^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2), n))}

Crossrefs

Cf. A029839, A185338, A208605.

Sequence in context: A034949 A263767 A185338 * A340988 A134013 A112301

Adjacent sequences: A208600 A208601 A208602 * A208604 A208605 A208606

Keyword

sign

Author

Michael Somos, Feb 29 2012

Status

approved