A131123 McKay-Thompson series of class 8A for the Monster group with a(0) = 8.

1, 8, 36, 128, 386, 1024, 2488, 5632, 12031, 24576, 48308, 91904, 170110, 307200, 542872, 941056, 1602819, 2686976, 4439688, 7238272, 11657090, 18561024, 29242240, 45617664, 70507772, 108036096, 164192188, 247620352, 370726652

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..996

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^-1 * (chi(-q^4) / chi(-q))^8 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^4) / ( eta(q) * eta(q^8) ))^8 in powers of q.

Euler transform of period 8 sequence [ 8, 0, 8, -8, 8, 0, 8, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * (1 - v*w) * (1 - v*u) - (v - u^2) * (v - w^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A007265(n) unless n=0.

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

T8A = 1/q + 8 + 36*q + 128*q^2 + 386*q^3 + 1024*q^4 + 2488*q^5 + 5632*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^4, q^4])^8, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(8*k))))^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)))^8, n))};

Crossrefs

Cf. A007265.

Sequence in context: A213581 A276279 A210379 * A055910 A022573 A014477

Adjacent sequences: A131120 A131121 A131122 * A131124 A131125 A131126

Keyword

nonn

Author

Michael Somos, Jun 15 2007

Status

approved