A186829 McKay-Thompson series of class 12A for the Monster group with a(0) = 6.

1, 6, 15, 32, 87, 192, 343, 672, 1290, 2176, 3705, 6336, 10214, 16320, 25905, 39936, 61227, 92928, 138160, 204576, 300756, 435328, 626727, 897408, 1271205, 1790592, 2508783, 3487424, 4824825, 6641664, 9083400, 12371904, 16778784, 22630912

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. A. Edgar, Table of n, a(n) for n = -1..1000

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.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 12 sequence [ 6, -6, 12, 0, 6, -12, 6, 0, 12, -6, 6, 0, ...].

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

Convolution inverse of A107653. Convolution square of A058571. Sixth convolution power of A112206.

G.f.: (1/x) * (Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(6*k - 3)))^6.

a(n) = -(-1)^n * A121666(n).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 02 2015

Expansion of eta(q^2)^12 * eta(q^6)^12 / (eta(q)^6 * eta(q^3)^6 * eta(q^4)^6 * eta(q^12)^6) in powers of q. - G. A. Edgar, Mar 11 2017

Example

G.f. = 1/q + 6 + 15*q + 32*q^2 + 87*q^3 + 192*q^4 + 343*q^5 + 672*q^6 + 1290*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q, q^2]  QPochhammer[ -q^3, q^6])^6, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^12 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^6, n))}
(PARI) q='q+O('q^50); Vec( eta(q^2)^12 * eta(q^6)^12 / (eta(q)^6 * eta(q^3)^6 * eta(q^4)^6 * eta(q^12)^6) ) \\ Joerg Arndt, Mar 11 2017

Crossrefs

Cf. A112147, A058571, A107653, A112206, A121666.

Sequence in context: A192747 A231264 A121666 * A231452 A118734 A200895

Adjacent sequences: A186826 A186827 A186828 * A186830 A186831 A186832

Keyword

nonn

Author

Michael Somos, Feb 27 2011

Status

approved