A152944 McKay-Thompson series of class 17A for the Monster group with a(0) = 2.

1, 2, 7, 14, 29, 50, 92, 148, 246, 386, 603, 904, 1367, 1996, 2914, 4160, 5924, 8290, 11581, 15942, 21878, 29712, 40184, 53876, 71979, 95436, 126097, 165556, 216594, 281848, 365548, 471808, 607050, 777794, 993528, 1264338, 1604434, 2029026

Offset

-1,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * ((psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)))^2 in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(-1) * (F(q) - q^4 / F(q))^2 / (chi(-q) * chi(-q^17))^4 in powers of q where F(q) = G(q^17) / G(q), G(q) = chi(q) * chi(-q^2) and chi() is a Ramanujan theta function.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (17 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058530(n) unless n = 0. Convolution square of A058639.

a(n) ~ exp(4*Pi*sqrt(n/17)) / (sqrt(2) * 17^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

Example

G.f. = 1/q + 2 + 7*q + 14*q^2 + 29*q^3 + 50*q^4 + 92*q^5 + 148*q^6 + 246*q^7 + ...

Mathematica

QP = QPochhammer; s = (QP[q^4]^2*(QP[q^34]^5/(QP[q]*QP[q^2]* QP[q^17]^3* QP[q^68]^2)) - q^4*QP[q^2]^5*(QP[q^68]^2/(QP[q]^3*QP[q^4]^2*QP[q^17]* QP[q^34])))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
nmax = 60; CoefficientList[Series[(Product[(1+x^k) * (1+x^(2*k))^2 * (1+x^(17*k)) * (1+x^(34*k-17))^2, {k, 1, nmax}] - x^4*Product[(1+x^k) * (1+x^(2*k-1))^2 * (1+x^(17*k)) * (1+x^(34*k))^2, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, May 01 2017 *)
a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^17] - EllipticTheta[ 2, 0, q^17] EllipticTheta[ 3, 0, q]) / (QPochhammer[ q] QPochhammer[ q^16]))^2 / (4 q^(3/2)), {q, 0, n}]; (* Michael Somos, Sep 06 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^17 + A)^3 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^17 + A) * eta(x^34 + A)))^2, n))};

Crossrefs

Cf. A058530, A058639.

Sequence in context: A221317 A005998 A122751 * A353215 A018437 A286857

Adjacent sequences: A152941 A152942 A152943 * A152945 A152946 A152947

Keyword

nonn

Author

Michael Somos, Dec 15 2008

Status

approved