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%I #17 Jun 17 2018 14:52:37
%S 1,1,11,20,57,92,207,312,623,932,1674,2464,4162,6024,9595,13748,21126,
%T 29820,44449,62004,90191,124288,177135,241632,338508,457272,631031,
%U 845008,1150752,1528380,2057700,2712192,3614217,4730148,6245541,8119672
%N McKay-Thompson series of class 14A for the Monster group with a(0) = 1.
%H G. C. Greubel, <a href="/A134782/b134782.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 (14)(7)(2)(1).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A058497(n) unless n=0. Convolution with A030187 is A028997.
%F a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (2^(3/4) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%F Expansion of A - 3 + 1/A, where A = (eta(q^2)*eta(q^7)/(eta(q)*eta(q^14) ))^4, in powers of q. - _G. C. Greubel_, Jun 17 2018
%e G.f. = 1/q + 1 + 11*q + 20*q^2 + 57*q^3 + 92*q^4 + 207*q^5 + 312*q^6 + 623*q^7 + ...
%t QP = QPochhammer; A = q^2*O[q]^40; A = (QP[q + A]*(QP[q^7 + A]/(QP[q^2 + A]*QP[q^14 + A])))^3/q; s = q*(4 + A + 8/A); CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)
%t a[ n_] := With[{A = (QPochhammer[ q] QPochhammer[ q^7] / (QPochhammer[ q^2] QPochhammer[ q^14]))^3 / q}, SeriesCoefficient[ 4 + A + 8 / A, {q, 0, n}]]; (* _Michael Somos_, May 05 2016 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; A:= (eta[q^2]*eta[q^7]/(eta[q]* eta[q^14]))^4; a:= CoefficientList[Series[q*(A - 3 + 1/A), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 17 2018 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)))^3 / x; polcoeff( (4 + A + 8 / A), n))};
%o (PARI) q='q+O('q^50); A = (eta(q^2)*eta(q^7)/(eta(q)*eta(q^14)))^4/q; Vec(A - 3 +1/A) \\ _G. C. Greubel_, Jun 17 2018
%Y Cf. A028997, A030187, A058497.
%K nonn
%O -1,3
%A _Michael Somos_, Nov 22 2007
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