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A123676 McKay-Thompson series of class 18C for the Monster group with a(0) = -3. 7
1, -3, 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, -23472, 26904, -30876 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

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 -4 + psi(q) / (q * psi(q^9)) + 3 * q * psi(q^9) / psi(q) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 09 2012

Expansion of (1/q) * (chi(-q) * chi(-q^9))^3 / chi(-q^3)^2 in powers of q where chi() is a Ramanjuan theta function.

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

Given g.f. A(x), then B(x) = 1/A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v - u * v * (6 + 4*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A123629.

a(n) = A058533(n) = A215412(n) = A215413(n) unless n=0. - Michael Somos, Aug 09 2012

Convolution inverse of A123629.

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2015

EXAMPLE

1/q - 3 + 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^9]/(QP[q^2]*QP[q^18])))^3*(QP[q^6]/ QP[q^3])^2 + O[q]^50; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 12 2015, adapted from PARI *)

nmax = 60; CoefficientList[Series[Product[(1+x^(3*k))^2 / ( (1+x^k)^3 * (1+x^(9*k))^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2015 *)

PROG

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

CROSSREFS

Cf. A058533, A123629, A215412, A215413.

Sequence in context: A268931 A003560 A262212 * A326814 A122775 A086632

Adjacent sequences:  A123673 A123674 A123675 * A123677 A123678 A123679

KEYWORD

sign

AUTHOR

Michael Somos, Oct 05 2006

STATUS

approved

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Last modified May 17 18:35 EDT 2021. Contains 343987 sequences. (Running on oeis4.)