A007252 - OEIS

%I M4599 #32 Jan 01 2017 09:56:20

%S 1,0,9,10,-30,6,-25,96,60,-250,45,-150,544,360,-1230,184,-675,2310,

%T 1410,-4830,750,-2450,8196,4920,-16180,2376,-7875,25644,15000,-48720,

%U 7126,-22800,73221,42310,-134760,19284,-61400,194334,110610,-349000,49563,-155250,486370

%N McKay-Thompson series of class 5B for the Monster group with a(0) = 0.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A007252/b007252.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 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 6 + (eta(q) / eta(q^5))^6 in powers of q. - _Michael Somos_, Apr 30 2004

%F a(n) = A045483(n) = A106248(n) unless n=0.

%F a(n) = A229793(n) - A078905(n) for n > 0. - _Seiichi Manyama_, Jan 01 2017

%e T5B = 1/q + 9*q + 10*q^2 - 30*q^3 + 6*q^4 - 25*q^5 + 96*q^6 + 60*q^7 - ...

%t a[ n_] := SeriesCoefficient[ 6 + 1/q (QPochhammer[ q] / QPochhammer[ q^5])^6, {q, 0, n}]; (* _Michael Somos_, May 22 2013 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 6*x + (eta(x + A) / eta(x^5 + A))^6, n))}; /* _Michael Somos_, Apr 30 2004 */

%o (PARI) {a(n) = my(A, k); if( n<-1, 0, k = (sqrtint(40*n + 48) + 7)\10; A = x * (sum(i=-k, k, (-1)^i * x^((5*i^2 + 3*i)/2), x^2 * O(x^n)) / sum(i=-k, k, (-1)^i * x^((5*i^2 + i)/2), x^2 * O(x^n)))^5; polcoeff( 1/A - A - 5, n))}; /* _Michael Somos_, Apr 30 2004 */

%Y Cf. A045483, A106248. (same except for initial terms).

%K sign

%O -1,3

%A _N. J. A. Sloane_