A133099 McKay-Thompson series of class 13B for the Monster group with a(0) = -2.

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, 0, 0, 4, -1, -4, 6, 2, 4, -6, -2, -4, -4, 0, -1, 0, 11, -2, -11, 16, 6, 12, -14, -2, -12, -12, 3, -2, -2, 24, -5, -24, 34, 12, 25, -32, -6, -24, -22, 4, -3, -2, 51, -10, -50, 72, 27, 52, -64, -12, -50, -48, 10, -8, -4, 98, -20, -96, 136, 48, 96, -120, -24, -90, -86, 16, -14, -8, 184, -36, -176, 252

Offset

-1,2

Links

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

Formula

Expansion of (eta(q) / eta(q^13))^2 in powers of q.

Euler transform of period 13 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - 13*u*v - 4*u*v * (u+v) - (u*v)^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - u*v + v^2)^2 - u*v * (13 + 6*u + u^2) * (13 + 6*v + v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (13 t)) = 13 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A121597.

G.f.: x^(-1) * (Product_{k>0} (1 - x^k) / (1 - x^(13*k)))^2.

a(n) = A034318(n) - 13*A121597(n). - Michael Somos, Jul 05 2012

a(n) = A058496(n) unless n=0. Convolution inverse of A121597.

a(n) = A214157(n) - A092876(n) unless n=0. - Michael Somos, Jan 17 2015

a(-1) = 1, a(n) = -(2/(n+1))*Sum_{k=1..n+1} A284587(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017

Example

G.f. = 1/q - 2 - q + 2*q^2 + q^3 + 2*q^4 - 2*q^5 - 2*q^7 - 2*q^8 + q^9 + ...

Mathematica

a[ n_] :=  SeriesCoefficient[ (1/q) (QPochhammer[ q] / QPochhammer[ q^13])^2, {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
a[ n_] := With[{A = q Product[ (1 - q^k)^KroneckerSymbol[ 13, k], {k, n + 1}]}, SeriesCoefficient[ 1/A - 3 - A, {q, 0, n}]]; (* Michael Somos, Jan 17 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^13 + A))^2, n))};

Crossrefs

Cf. A034381, A058496, A092876, A121597, A214157.

Sequence in context: A002107 A208845 A232506 * A006571 A243906 A100889

Adjacent sequences: A133096 A133097 A133098 * A133100 A133101 A133102

Keyword

sign

Author

Michael Somos, Sep 11 2007

Status

approved