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%I #21 Jun 17 2018 16:03:53
%S 1,1,8,22,42,70,155,246,421,722,1101,1730,2761,4062,6106,9040,13065,
%T 18806,27081,37950,53183,74290,102213,140048,191612,258426,348300,
%U 467484,622023,825016,1090957,1432290,1875930,2448610,3179136,4114996
%N McKay-Thompson series of class 15A for the Monster group with a(0) = 1.
%H G. C. Greubel, <a href="/A134783/b134783.txt">Table of n, a(n) for n = -1..1000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Associated with permutations in Mathieu group M24 of shape (15)(5)(3)(1).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A058508(n) unless n=0. Convolution with A030184 is A028998.
%F a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%F Expansion of A + 3 + 9/A, where A = (eta(q)*eta(q^5)/(eta(q^3)*eta(q^15)) ))^2, in powers of q. - _G. C. Greubel_, Jun 17 2018
%e G.f. = 1/q + 1 + 8*q + 22*q^2 + 42*q^3 + 70*q^4 + 155*q^5 + 246*q^6 + 421*q^7 + ...
%t QP = QPochhammer; A = q^2*O[q]^40; A = (QP[q + A]*(QP[q^5 + A]/(QP[q^3 + A]*QP[q^15 + A])))^2/q; s = q*(3 + A + 9/A); CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)
%t a[ n_] := With[{A = (QPochhammer[ q^3] QPochhammer[ q^5] / (QPochhammer[ q] QPochhammer[ q^15]))^3 /q}, SeriesCoefficient[ -2 + A - 1/A, {q, 0, n}]]; (* _Michael Somos_, May 05 2016 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) * eta(x^5 + A) / (eta(x^3 + A) * eta(x^15 + A)))^2 / x; polcoeff( (3 + A + 9 / A), n))}
%Y Cf. A028998, A030184, A058498.
%K nonn
%O -1,3
%A _Michael Somos_, Nov 22 2007
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