A215413 McKay-Thompson series of class 18C for the Monster group with a(0) = 1.

1, 1, 3, -2, 3, -6, 10, -12, 15, -22, 30, -36, 44, -60, 78, -96, 117, -150, 190, -228, 276, -340, 420, -504, 603, -732, 885, -1052, 1245, -1488, 1770, -2088, 2454, -2902, 3420, -3996, 4666, -5460, 6378, -7400, 8583, -9972, 11566, -13344, 15378, -17752, 20448

Offset

-1,3

Comments

A058533, A123676, A215412, A058644, A215413 are all essentially the same sequence. - N. J. A. Sloane, Aug 09 2012

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 18C.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(q) / (q * psi(q^9)) + 3 * q * psi(q^9) / psi(q) in powers of q where psi() is a Ramanujan theta function.

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

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

Euler transform of period 18 sequence [ 1, 2, -5, 2, 1, -2, 1, 2, -4, 2, 1, -2, 1, 2, -5, 2, 1, 0, ...].

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A123629.

a(n) = A058533(n) = A123676(n) = A215412(n) unless n=0.

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

Example

1/q + 1 + 3*q - 2*q^2 + 3*q^3 - 6*q^4 + 10*q^5 - 12*q^6 + 15*q^7 - 22*q^8 + ...

Mathematica

QP = QPochhammer; s = QP[q^3]^6 / (QP[q] * QP[q^2] * QP[q^6]^2 * QP[q^9] * QP[q^18]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 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)^6 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^18 + A)), n))}

Crossrefs

Cf. A058533, A123629, A123676, A215412.

Sequence in context: A274462 A050062 A058533 * A058644 A049923 A184881

Adjacent sequences: A215410 A215411 A215412 * A215414 A215415 A215416

Keyword

sign

Author

Michael Somos, Aug 09 2012

Status

approved