A093073 McKay-Thompson series of class 18A for the Monster group with a(0) = -1.

1, -1, -2, 1, 0, 2, 1, 0, 0, -1, 0, -4, -1, 0, 4, 0, 0, 2, 1, 0, -8, 2, 0, 8, 0, 0, 2, -2, 0, -16, -3, 0, 16, -1, 0, 4, 4, 0, -28, 4, 0, 28, 1, 0, 8, -4, 0, -48, -6, 0, 46, -1, 0, 12, 5, 0, -80, 8, 0, 76, 1, 0, 20, -8, 0, -126, -10, 0, 120, -2, 0, 32, 11, 0, -196, 14, 0, 184, 4, 0, 48

Offset

-1,3

Links

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

Formula

Expansion of eta(q) * eta(q^2) / (eta(q^9) * eta(q^18)) in powers of q.

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -1, 0, ...]. - Corrected by Sean A. Irvine, Mar 06 2020

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^4 + v^4 - u*v * ((u + v)^2 + 9*(u + v) + u*v * (u + v + 4)).

a(3*n - 1) = A062242(n), a(3*n + 1) = -2*A092848(n). a(3*n) = 0, unless n=0. a(n) = A058531(n) unless n=0.

Example

G.f. = 1/q - 1 - 2*q + q^2 + 2*q^4 + q^5 - q^8 - 4*q^10 - q^11 + 4*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^(-1) QPochhammer[ q] QPochhammer[ q^2] / (QPochhammer[ q^9] QPochhammer[ q^18]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x + x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) / (eta(x^9 + A) * eta(x^18 + A)), n))}

Crossrefs

Cf. A062242, A092848, A058531.

Sequence in context: A067613 A264034 A058531 * A251635 A156319 A190893

Adjacent sequences: A093070 A093071 A093072 * A093074 A093075 A093076

Keyword

sign

Author

Michael Somos, Mar 17 2004

Extensions

Edited for better readability and coherence. - Michael Somos, Oct 23 2013

Status

approved