A215412 McKay-Thompson series of class 18C for the Monster group with a(0) = -2.

1, -2, 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

-1,2

Comments

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

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

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 -3 + 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 (1/q) * (psi(q^3)^2 / (psi(q) * psi(q^9)))^2 in powers of q where psi() is a Ramanujan theta function.

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

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227587. - Michael Somos, Jul 16 2013

a(n) = A058533(n) = A123676(n) = A215413(n) unless n=0.

a(n) = -(-1)^n * A227585(n). - Michael Somos, Jul 16 2013

Convolution square of A112176. - Michael Somos, Jul 16 2013

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

Example

1/q - 2 + 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] * QP[q^6]^4 * QP[q^9])^2 / (QP[q^2] * QP[q^3] * QP[q^18])^4 + 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 + A) * eta(x^6 + A)^4 * eta(x^9 + A))^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^18 + A))^4, n))}

Crossrefs

Cf. A058533, A112176, A123676, A215413, A227585, A227587.

Sequence in context: A336953 A193917 A089135 * A227585 A038063 A264506

Adjacent sequences: A215409 A215410 A215411 * A215413 A215414 A215415

Keyword

sign

Author

Michael Somos, Aug 09 2012

Status

approved