A007260 McKay-Thompson series of class 6a for Monster.

1, -33, -153, -713, -2550, -7479, -20314, -51951, -122229, -276656, -601068, -1254105, -2541531, -5018721, -9647991, -18168984, -33554784, -60818040, -108471674, -190607871, -330140403, -564580142, -953980392, -1593599832, -2634301308, -4311874755, -6991318008

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

A more correct name would be: Expansion of replicable function of class 6a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015

Examples

			G.f. = 1 - 33*x - 153*x^2 - 713*x^3 - 2550*x^4 - 7479*x^5 - 20314*x^6 + ...
T6a = 1/q - 33*q - 153*q^3 - 713*q^5 - 2550*q^7 - 7479*q^9 - 20314*q^11 + ...
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..499 from G. A. Edgar)
• D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
• J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
• J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A007242, A007250, A007262, A121596, A258917.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^3])^6 - 27 x (QPochhammer[ x^3] / QPochhammer[ x])^6, {x, 0, n} ]; (* Michael Somos, Jun 14 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^6 - 27 * x * (eta(x^3 + A) / eta(x + A))^6, n))}; /* Michael Somos, Jun 14 2015 */

PARI

{ my(q='q+O('q^66), t=(eta(q)/eta(q^3))^6 ); Vec( t - 27*q/t ) } \\ Joerg Arndt, Apr 02 2017

Formula

Expansion of q * ((eta(q^2) / eta(q^6))^6 - 27 * (eta(q^6) / eta(q^2))^6) in powers of q^2. - Michael Somos, Jun 14 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -1 / f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 14 2015

a(n) = A007262(n) - 27 * A121596(n-1). - Michael Somos, Jun 14 2015

Convolution square is A258917. - Michael Somos, Jun 14 2015

a(n) ~ -exp(2*Pi*sqrt(2*n/3)) / (2^(3/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017