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A058532 McKay-Thompson series of class 18B for the Monster group. 3
1, 0, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

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).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

FORMULA

From Michael Somos, Aug 19 2012: (Start)

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

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

Expansion of -5 + (eta(q^3)^8 + 4 * eta(q^6)^8) /(eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^6)^2 * eta(q^9) * eta(q^18)).

a(n) = A215407(n) unless n=0. (End)

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

EXAMPLE

T18B = 1/q + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...

MATHEMATICA

QP = QPochhammer; s = -2*q+(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-Fran├žois Alcover, Nov 13 2015, after 3rd formula *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x + A) * eta(x^2 + A) / (eta(x^9 + A) * eta(x^18 + A)); polcoeff( x + A + 9 * x^2 / A, n))} /* Michael Somos, Aug 19 2012 */

CROSSREFS

Cf. A000521, A007240, A007241, A007267, A014708, A045478.

Cf. A093073, A123676, A128517, A215407.

Sequence in context: A064948 A064950 A240795 * A280966 A174466 A070422

Adjacent sequences:  A058529 A058530 A058531 * A058533 A058534 A058535

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

STATUS

approved

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Last modified January 21 11:21 EST 2019. Contains 319354 sequences. (Running on oeis4.)