%I #40 Oct 14 2018 04:42:43
%S 1,-4,2,8,-5,-4,-10,12,-7,8,46,-36,-26,-44,46,-28,42,188,-132,-96,
%T -167,172,-98,120,596,-420,-286,-492,496,-280,368,1680,-1151,-792,
%U -1332,1320,-735,916,4264,-2908,-1960,-3252,3200,-1764,2230,10104,-6798,-4560,-7536
%N Expansion of (eta(q) / eta(q^7))^4 in powers of q.
%C McKay-Thompson series of class 7B for the Monster group with a(0) = -4.
%D N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)
%H Seiichi Manyama, <a href="/A030181/b030181.txt">Table of n, a(n) for n = -1..10000</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. Elkies, <a href="http://www.msri.org/publications/books/Book35/files/elkies.pdf">The Klein quartic in number theory</a>
%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. 66.
%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 Euler transform of period 7 sequence [ -4, -4, -4, -4, -4, -4, 0, ...]. - _Michael Somos_, Mar 15 2004
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u-v)^2 * (u+v) - u*v * (u+7) * (v+7). - _Michael Somos_, Feb 19 2007
%F a(n) = A052240(n) unless n = 0.
%F a(-1) = 1, a(n) = -(4/(n+1))*Sum_{k=1..n+1} A113957(k)*a(n-k) for n > -1. - _Seiichi Manyama_, Mar 29 2017
%F a(n) = A108481(n) - A305443(n) - A262933(n) for n > 0. - _Seiichi Manyama_, Oct 14 2018
%e 1/q - 4 + 2*q + 8*q^2 - 5*q^3 - 4*q^4 - 10*q^5 + 12*q^6 - 7*q^7 + 8*q^8 + ...
%t QP = QPochhammer; s = (QP[q]/QP[q^7])^4 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^7 + A))^4, n))} /* _Michael Somos_, Feb 19 2007 */
%Y Cf. A052240, A108481, A262933, A305443.
%K sign
%O -1,2
%A _N. J. A. Sloane_