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%I M5111 #57 Nov 27 2017 03:24:54
%S 1,21,171,745,2418,7587,20510,51351,122715,277384,598812,1255761,
%T 2543973,5011725,9653013,18176040,33535032,60831648,108490390,
%U 190557015,330174837,564626278,953857104,1593681480,2634409140,4311592119,6991502688,11237020682,17909802270
%N McKay-Thompson series of class 6b for the Monster group.
%C From _Gary W. Adamson_, Jul 21 2009: (Start)
%C (1 + 21x + 171x^2 + 745x^3 + ...)^2 = (1 + 42x + 783x^2 + 8672x^3 + ...)
%C where A030197 = (1, 42, 783, 8672, 65367, ...). (End)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A007261/b007261.txt">Table of n, a(n) for n = 0..3000</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 (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/2) in powers of x where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jun 16 2012
%F Convolution cube of A058537. - _Michael Somos_, Aug 20 2012
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2015
%F Expansion of q^(1/2) * (eta(q)^6/eta(q^3)^6 + 27*eta(q^3)^6/eta(q)^6) in powers of q. - _G. A. Edgar_, Mar 10 2017
%F a(n) = A007262(n) + 27 * A121596(n-1). - _Sean A. Irvine_, Nov 26 2017
%e 1 + 21*x + 171*x^2 + 745*x^3 + 2418*x^4 + 7587*x^5 + 20510*x^6 + 51351*x^7 + ...
%e T6b = 1/q + 21*q + 171*q^3 + 745*q^5 + 2418*q^7 + 7587*q^9 + 20510*q^11 + ...
%t a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[Sqrt[(1 + 27*x*A)^2/A], n]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 06 2015, adapted from _Michael Somos_'s PARI script *)
%t CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^3 / (QPochhammer[x, x]^3*QPochhammer[x^3, x^3]^6), {x, 0, 50}], x] (* _Vaclav Kotesovec_, Nov 07 2015 *)
%t nmax = 30; CoefficientList[Series[Product[(1 - x^k)^6/(1 - x^(3*k))^6, {k, 1, nmax}] + 27*x*Product[(1 - x^(3*k))^6/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 11 2017 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( sqrt((1 + 27 * x * A)^2 / A), n))} /* _Michael Somos_, Jun 16 2012 */
%o (PARI) N=66; q='q+O('q^N); t=(eta(q) / eta(q^3))^6; Vec(t + 27*q/t) \\ _Joerg Arndt_, Mar 11 2017
%Y Cf. A030197. - _Gary W. Adamson_, Jul 21 2009
%Y Cf. A058537.
%K nonn
%O 0,2
%A _N. J. A. Sloane_