A215407 McKay-Thompson series of class 18B for the Monster group with a(0) = 2.

1, 2, 7, 10, 27, 38, 82, 108, 207, 278, 486, 644, 1052, 1404, 2182, 2880, 4293, 5654, 8182, 10692, 15076, 19604, 27108, 35000, 47547, 61020, 81713, 104236, 137781, 174800, 228498, 288360, 373174, 468566, 601020, 751036, 955642, 1188756, 1501730, 1859944

Offset

-1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). See Table 4 18B.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * psi(q^3)^2 / (psi(q) * phi(q^9)) * (f(-q^3)^2 / (f(-q) * f(-q^9)))^3 in powers of q where psi(), f() are Ramanujan theta functions.

Expansion of ((c(q) * c(q^2) * b(q^3) * b(q^6)) / (b(q) * b(q^2) * c(q^3) * c(q^6)))^(1/2) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q^3) * eta(q^6))^4 / (eta(q) * eta(q^2) * eta(q^9) * eta(q^18))^2 in powers of q.

Euler transform of period 18 sequence [ 2, 4, -2, 4, 2, -4, 2, 4, 0, 4, 2, -4, 2, 4, -2, 4, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * v * (u + 3) * (v + 3) * (u + v) - (u^2 + 4 * u * v + v^2)^2.

G.f. A(x) = B(x) * B(x^2) where B(x) is g.f. for A058601.

a(n) = A058532(n) unless n=0. Convolution square of A058646.

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

Example

1/q + 2 + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(((1-x^(3*k)) * (1-x^(6*k)))^2 / ((1-x^k) * (1-x^(2*k)) * (1-x^(9*k)) * (1-x^(18*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s=(QP[q^3]*QP[q^6])^4/(QP[q]*QP[q^2]*QP[q^9]*QP[q^18])^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^2 + A) * eta(x^9 + A) * eta(x^18 + A))^2, n))}

Crossrefs

Cf. A058532, A058601, A058646.

Sequence in context: A032007 A091295 A361931 * A084184 A015963 A293002

Adjacent sequences: A215404 A215405 A215406 * A215408 A215409 A215410

Keyword

nonn

Author

Michael Somos, Aug 09 2012

Status

approved