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%I #31 Mar 12 2021 22:24:42
%S 1,26,79,326,755,2106,4460,10284,20165,41640,77352,147902,263019,
%T 475516,816065,1413142,2353446,3936754,6391091,10390150,16497734,
%U 26184098,40775677,63394792,97037170,148178934,223351867,335704742,499050461,739575640,1085723797
%N McKay-Thompson series of class 8C for Monster.
%C Let q = exp(-Pi*sqrt(58)/4). Then 396 = B(q) = 1/q + 26*q^3 + ... + a(n)*q^(4*n-1) + ... - _Michael Somos_, Sep 30 2019
%H G. C. Greubel, <a href="/A052241/b052241.txt">Table of n, a(n) for n = 0..1000</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 Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of 2 * q^(1/4) * ((k'^4 + 4*k^2) / (k'^2 * k))^(1/2) in powers of q. - _Michael Somos_, Sep 01 2014
%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - (u*v - 12) * (u*v - 32)^2. - _Michael Somos_, Sep 01 2014
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 01 2014
%F Convolution square is A007247. Convolution fourth power is A007267.
%F a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, May 01 2017
%e G.f. = 1 + 26*x + 79*x^2 + 326*x^3 + 755*x^4 + 2106*x^5 + 4460*x^6 + ...
%e T8C = 1/q + 26*q^3 + 79*q^7 + 326*q^11 + 755*q^15 + 2106*q^19 + 4460*q^23 + ...
%t QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^2 + A])^12; s = Sqrt[A + 64*(q/A)]; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 13 2015, adapted from PARI *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; e4D := q^(1/2)*(eta[q]/eta[q^2])^12;
%t T4B := e4D + 64*q/e4D; a[n_]:= SeriesCoefficient[Sqrt[(T4B /. {q -> q^2}) + O[q]^100], {q, 0, n}]; Table[a[n], {n, 0, 50}][[1 ;; ;; 2]] (* _G. C. Greubel_,Feb 13 2018 *)
%t a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, A}, A = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (A + 64/A)^(1/2), {q, 0, n - 1/4}]]; (* _Michael Somos_, Sep 30 2019 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( sqrt(A + 64 * x / A), n))}; /* _Michael Somos_, Sep 01 2014 */
%Y Cf. A007247, A007249, A007267.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000
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