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%I #35 Mar 13 2019 21:06:40
%S 1,14,65,156,456,1066,2250,4720,9426,17590,32801,58904,102650,176646,
%T 298066,491792,803923,1293450,2051156,3221716,5004028,7682744,
%U 11703580,17663312,26423351,39248618,57866503,84685920,123188502,178054416,255782770,365467216
%N McKay-Thompson series of class 9a for the Monster group.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C In volume 2 of Raamunjuan's Notebooks is an obscure equation involving t(1-t) on the left and GG' on the right and they both are equal to the g.f. of 1/3 of this sequence. Here t^(1/3) = c(x)/a(x), (1-t)^(1/3) = b(x)/a(x) since a(x)^3 = b(x)^3 + c(x)^3. N.B. The left side was (t(1-t))^(1/3) but the exponent should be (-1/3) instead which is why the equation was so obscure. - _Michael Somos_, Mar 13 2019
%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179.
%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.
%H Vaclav Kotesovec, <a href="/A058092/b058092.txt">Table of n, a(n) for n = 0..2000</a>
%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 G. Manco, <a href="https://math.stackexchange.com/q/2104356">How to calculate moduli alpha_3n of the Ramanujan's q_3 theory</a>, Mathematics StackExchange, Jan 2017.
%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)^(1/3) in powers of x where b(), c() are cubic AGM theta functions, _Michael Somos_, Jun 16 2012
%F Convolution cube is A030197.
%F a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2015
%e G.f. = 1 + 14*x + 65*x^2 + 156*x^3 + 456*x^4 + 1066*x^5 + 2250*x^6 + 4720*x^7 + ...
%e T9a = 1/q + 14*q^2 + 65*q^5 + 156*q^8 + 456*q^11 + 1066*q^14 + 2250*q^17 + ...
%t a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[((1 + 27*x*A)^2/A)^(1/3), n]]; Table[a[n], {n, 0, 40}] (* _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)^2 / (QPochhammer[x, x]^2*QPochhammer[x^3, x^3]^4), {x, 0, 50}], x] (* _Vaclav Kotesovec_, Nov 07 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/3), n))}; /* _Michael Somos_, Jun 16 2012 */
%Y Cf. A030197, A051273.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000
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