A226009 McKay-Thompson series of class 33A for the Monster group with a(0) = -1.

1, -1, -1, 1, -1, 0, 2, -1, -1, 3, -2, -2, 5, -2, -3, 6, -4, -4, 9, -5, -7, 12, -7, -7, 18, -9, -10, 22, -13, -14, 31, -16, -18, 39, -22, -24, 53, -28, -31, 66, -37, -38, 87, -46, -51, 108, -59, -64, 138, -74, -80, 171, -94, -100, 216, -115, -126, 266, -144

Offset

-1,7

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Formula

Expansion of eta(q) * eta(q^11) / (eta(q^3) * eta(q^33)) in powers of q.

Euler transform of period 33 sequence [ -1, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 0, -1, -1, 0, -1, -1, 0, -1, -1, 0, -2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 3) - (u+v) * (u^2 - 3 * u*v + v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (33 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128663.

G.f.: 1/x * Product_{k>0} (1 - x^k) * (1 - x^(11*k)) / ((1 - x^(3*k)) * (1 - x^(33*k))).

a(n) = A058636(n) unless n=0. Convolution inverse of A128663.

Example

1/q - 1 - q + q^2 - q^3 + 2*q^5 - q^6 - q^7 + 3*q^8 - 2*q^9 - 2*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^11] / (q QPochhammer[ q^3] QPochhammer[ q^33]), {q, 0, n}] (* Michael Somos, Jul 25 2013 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A) / (eta(x^3 + A) * eta(x^33 + A)), n))}

Crossrefs

Cf. A058636, A128663.

Sequence in context: A094340 A228668 A058636 * A132462 A161039 A104467

Adjacent sequences: A226006 A226007 A226008 * A226010 A226011 A226012

Keyword

sign

Author

Michael Somos, May 22 2013

Status

approved