A143751 McKay-Thompson series of class 60D for the Monster group with a(0) = -1.

1, -1, -1, 1, 0, 0, 0, -1, 1, 1, -1, -1, 1, 0, -1, 2, 0, -2, 2, -1, 0, 2, -4, 0, 5, -1, -4, 2, 1, -2, 3, -3, -2, 7, -5, -2, 8, -6, -5, 8, 1, -5, 2, -2, -1, 12, -11, -10, 21, -6, -10, 13, -7, -4, 11, -7, -4, 14, -13, -10, 33, -14, -28, 32, -3, -12, 18, -24, 1, 36, -27, -22, 44, -13, -35, 50, -13, -36, 46, -26, -6, 56, -63, -22, 89, -30

Offset

-1,16

Links

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

Michael Somos, A Remarkable eta-product Identity

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of eta(q) * eta(q^12) * eta(q^15) * eta(q^20) / (eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60)) in powers of q.

Expansion of F(q) * F(q^2) in powers of q^3 where F(q) is the g.f. of A112215.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A143752.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 + v^2) * (1 + u + v) * (u + v + u*v) - u*v * (1 + 2*u + 2*v + u*v)^2.

G.f.: (x * Product_{k>0} P(30, x^k) * P(60, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.

A058728(n) = a(n) unless n=0. Convolution inverse of A143752.

Example

G.f. = 1/q - 1 - q + q^2 - q^6 + q^7 + q^8 - q^9 - q^10 + q^11 - q^13 + 2*q^14 + ...

Mathematica

QP = QPochhammer; s = QP[q]*QP[q^12]*QP[q^15]*(QP[q^20]/(QP[q^3]*QP[q^4]* QP[q^5]*QP[q^60])) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A) / (eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A)), n))};
(Magma) S<x> := PowerSeriesRing(RationalField()); Coefficients( DedekindEta(x)*DedekindEta(x^12)*DedekindEta(x^15)*DedekindEta(x^20)/( DedekindEta(x^3) *DedekindEta(x^4)*DedekindEta(x^5)*DedekindEta(x^60))); // G. C. Greubel, Mar 04 2018

Crossrefs

Cf. A058728, A112215, A143752.

Sequence in context: A099751 A159937 A058728 * A353322 A158950 A213013

Adjacent sequences: A143748 A143749 A143750 * A143752 A143753 A143754

Keyword

sign

Author

Michael Somos, Aug 31 2008

Status

approved