A058618 McKay-Thompson series of class 30G for the Monster group.

1, 0, 1, -1, 2, -2, 2, -3, 5, -5, 5, -7, 9, -10, 11, -14, 18, -20, 22, -27, 32, -36, 40, -48, 57, -63, 70, -82, 95, -106, 119, -137, 158, -175, 195, -222, 252, -280, 311, -352, 397, -439, 486, -546, 611, -676, 747, -834, 929, -1024, 1128, -1253, 1389, -1528

Offset

-1,5

References

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Links

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Euler transform of period 30 sequence [ 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0, ...]. - Michael Somos, Apr 06 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A094023.

G.f.: (x * Product_{k>0} (1 + x^k) * (1 + x^(15*k)) * P(15, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.

Convolution inverse of A094022. a(2*n) = -A123630(n).

a(2*n) = -A094023(n) if n>0. - Michael Somos, Aug 26 2015

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

T30G = 1/q + q - q^2 + 2*q^3 - 2*q^4 + 2*q^5 - 3*q^6 + 5*q^7 - 5*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^3] QPochhammer[ q^5] / (QPochhammer[ q^2] QPochhammer[ q^30]), {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^30 + A)), n))}; /* Michael Somos, Apr 06 2012 */

Crossrefs

Cf. A094022, A094023, A123630.

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Sequence in context: A091974 A029073 A376812 * A135213 A145725 A058727

Adjacent sequences: A058615 A058616 A058617 * A058619 A058620 A058621

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved