A255391 McKay-Thompson series of class 24h for the Monster group with a(0) = -2.

1, -2, 1, 0, -1, 0, 1, 0, 2, 0, -1, 0, -2, 0, -1, 0, 3, 0, 0, 0, -4, 0, 1, 0, 5, 0, 1, 0, -7, 0, 0, 0, 8, 0, 0, 0, -10, 0, -1, 0, 13, 0, -2, 0, -16, 0, 0, 0, 20, 0, 3, 0, -24, 0, 2, 0, 30, 0, -2, 0, -36, 0, -4, 0, 43, 0, 0, 0, -52, 0, 3, 0, 61, 0, 2, 0, -73

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 (1/q) * chi(-q)^2 * chi(-q^4) * chi(q^3)^2 * chi(-q^12) in powers of q where chi() is a Ramanujan theta function.

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

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

a(2*n) = 0 unless n=0. a(2*n - 1) = A112165(n). a(n) = A255396(n) unless n=0.

Example

G.f. = 1/q - 2 + q - q^3 + q^5 + 2*q^7 - q^9 - 2*q^11 - q^13 + 3*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ q, q^2]^2 QPochhammer[ q^4, q^8] QPochhammer[ -q^3, q^6]^2 QPochhammer[ q^12, q^24], {q, 0, n}];

Prog

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

Crossrefs

Cf. A112165, A255396.

Sequence in context: A330935 A208769 A255327 * A255396 A116683 A079748

Adjacent sequences: A255388 A255389 A255390 * A255392 A255393 A255394

Keyword

sign

Author

Michael Somos, Feb 22 2015

Status

approved