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A045481 McKay-Thompson series of class 3B for the Monster group with a(0) = -3. 4

%I

%S 1,-3,54,-76,-243,1188,-1384,-2916,11934,-11580,-21870,79704,-71022,

%T -123444,421308,-352544,-581013,1885572,-1510236,-2388204,7469928,

%U -5777672,-8852004,26869968,-20218587,-30177684,89408826

%N McKay-Thompson series of class 3B for the Monster group with a(0) = -3.

%H Vaclav Kotesovec, <a href="/A045481/b045481.txt">Table of n, a(n) for n = -1..1000</a>

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 38.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of 9 + (eta(q) / eta(q^3))^12 in powers of q.

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

%t a[ n_] := With[{m = n + 1}, SeriesCoefficient[ 9 q + (Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 3, m, 3}])^12, {q, 0, m}]] (* _Michael Somos_, Nov 08 2011 *)

%t QP = QPochhammer; s = 9*q+(QP[q]/QP[q^3])^12 + O[q]^30; CoefficientList[s, q] (* _Jean-Fran├žois Alcover_, Nov 12 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 9*x + (eta(x + A) / eta(x^3 + A))^12, n))}; /* _Michael Somos_, Nov 08 2011 */

%Y Essentially same as A007244, A030182, A045481.

%K sign,easy,nice

%O -1,2

%A _N. J. A. Sloane_.

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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)