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%I #19 Sep 06 2018 18:34:01
%S 1,3,2,1,2,3,4,5,7,8,11,13,16,19,25,28,35,41,50,58,71,81,98,113,134,
%T 154,183,208,244,280,326,371,431,489,565,641,735,832,953,1075,1225,
%U 1382,1569,1764,1999,2243,2533,2839,3195,3575,4018,4484,5026,5604,6267,6975
%N Coefficients of replicable function number 49a with a(0) = 3.
%C Denoted 49Z in Conway+Norton with a slight typo in the formula on page 337.
%H Vaclav Kotesovec, <a href="/A136560/b136560.txt">Table of n, a(n) for n = -1..9998</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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of ( eta(q^7)^4 + 7 * eta(q)^2 * eta(q^49)^2 ) / ( eta(q) * eta(q^49) * ( eta(q)^2 + 7 * eta(q) * eta(q^49) + 7 * eta(q^49)^2 ) ) in powers of q.
%F a(n) ~ exp(4*Pi*sqrt(n)/7) / (sqrt(14)*n^(3/4)). - _Vaclav Kotesovec_, Dec 04 2015
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 06 2018
%e G.f. = 1/q + 3 + 2*q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 5*q^6 + 7*q^7 + 8*q^8 + ...
%t QP = QPochhammer; A1 = QP[q]; A2 = QP[q^7]; A3 = QP[q^49]; s = (A2^4 + 7*q^3*A1^2*A3^2)/(A1*A3)/(A1^2 + 7*q^2*A1*A3 + 7*q^4*A3^2) + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%t a[ n_] := With[ {e1 = QPochhammer[ q], e2 = QPochhammer[ q^7], e3 = QPochhammer[ q^49]}, SeriesCoefficient[ (e2^4 + 7 q^3 e1^2 e3^2) / (q e1 e3 (e1^2 + 7 q^2 e1 e3 + 7 q^4 e3^2)), {q, 0, n}]]; (* _Michael Somos_, Sep 06 2018 *)
%o (PARI) {a(n) = my(A, A1, A2, A3); if( n<-1, 0, n++; A = x * O(x^n); A1 = eta(x + A); A2 = eta(x^7 + A); A3 = eta(x^49 + A); polcoeff( (A2^4 + 7 * x^3 * A1^2 * A3^2) / (A1 * A3) / (A1^2 + 7 * x^2 * A1*A3 + 7 * x^4 * A3^2 ), n))};
%Y Cf. A058700(n) = a(n) unless n=0.
%K nonn
%O -1,2
%A _Michael Somos_, Jan 05 2008
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