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A030182 McKay-Thompson series of class 3B for the Monster group with a(0) = -12. 6
1, -12, 54, -76, -243, 1188, -1384, -2916, 11934, -11580, -21870, 79704, -71022, -123444, 421308, -352544, -581013, 1885572, -1510236, -2388204, 7469928, -5777672, -8852004, 26869968, -20218587, -30177684, 89408826 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

COMMENTS

Let t(q) = (eta(q)/eta(q^3))^12 = 1/q-12+54q-76q^2-243q^3+.... If j(q) is the j-invariant, with q-series given by A000521, then j(q) = (t+27)(t+243)^3/t^3 j(q^3) = (t+27)(t+3)^3/t. Hence t(q) can be used to parametrize the classical modular curve X0(3). - Gene Ward Smith, Aug 04 2006

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

Seiichi Manyama, Table of n, a(n) for n = -1..10000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 38.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of (eta(q) / eta(q^3))^12 in powers of q.

Expansion of (3 * b(q) / c(q))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012

Euler transform of period 3 sequence [ -12, -12, 0, ...]. - Michael Somos, Nov 08 2011

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u + v)^3 - u * (27 + u) * v * (27 + v). - Michael Somos, Nov 08 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121590. - Michael Somos, Nov 08 2011

G.f.: x^-1 * (Product_{k>0} (1 - x^k) / (1 - x^(3*k)))^12.

Convolution inverse of A121590. Convolution square of A007262. Convolution cube of A058095. Convolution fourth power of A199659. Convolution sixth power of A112157. Convolution twelfth power of A137569.

a(-1) = 1, a(n) = -(12/(n+1))*Sum_{k=1..n+1} A046913(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017

EXAMPLE

G.f. = 1/q - 12 + 54*q - 76*q^2 - 243*q^3 + 1188*q^4 - 1384*q^5 - 2916*q^6 + ...

MATHEMATICA

a[ n_] := With[{m = n + 1}, SeriesCoefficient[ (Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 3, m, 3}])^12, {q, 0, m}]]; (* Michael Somos, Nov 08 2011 *)

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^3])^12, {q, 0, n}]; (* Michael Somos, May 03 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^12, n))}; /* Michael Somos, Nov 08 2011 */

CROSSREFS

Cf. A007244, A007262, A045481, A058095, A112157, A121590, A137569, A198955, A199659.

Sequence in context: A213549 A211060 A242514 * A060171 A133078 A034436

Adjacent sequences:  A030179 A030180 A030181 * A030183 A030184 A030185

KEYWORD

sign,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 17 05:26 EST 2019. Contains 319207 sequences. (Running on oeis4.)