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McKay-Thompson series of class 7A for the Monster group with a(0) = 9.
2

%I #19 Oct 26 2018 23:06:55

%S 1,9,51,204,681,1956,5135,12360,28119,60572,125682,251040,487426,

%T 920568,1699611,3070508,5445510,9490116,16283793,27537708,45959775,

%U 75760640,123471327,199081632,317814988,502608456,787889775,1224834672,1889206080,2892264900

%N McKay-Thompson series of class 7A for the Monster group with a(0) = 9.

%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 176 Entry 32(iii).

%H G. C. Greubel, <a href="/A282877/b282877.txt">Table of n, a(n) for n = -1..1000</a>

%F Expansion of (eta(q)/eta(q^7))^4 + 13 + 49*(eta(q^7)/eta(q))^4 in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = f(t) where q = exp(2 Pi i t).

%F Convolution cube of A058576.

%F a(n) ~ exp(4*Pi*sqrt(n/7)) / (sqrt(2)*7^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Feb 26 2017

%e G.f. = 1/q + 9 + 51*q + 204*q^2 + 681*q^3 + 1956*q^4 + 5135*q^5 + 12360*q^6 + ...

%t a[ n_] := With[ {A = q (QPochhammer[ q^7] / QPochhammer[ q])^4}, SeriesCoefficient[ 1/A + 13 + 49 A, {q, 0, n}]];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( 1/A + 13*x + 49*x^2 * A, n))};

%Y Cf. A007264, A030183, A045489, A058576.

%K nonn

%O -1,2

%A _Michael Somos_, Feb 23 2017