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%I M4131 #38 Jan 24 2023 17:36:01
%S 1,0,-6,20,15,36,0,-84,195,100,240,0,-461,1020,540,1144,0,-1980,4170,
%T 2040,4275,0,-6984,14340,6940,14076,0,-21936,44025,20760,41476,0,
%U -62484,123620,57630,113244,0,-166056,324120,148900,289578,0
%N McKay-Thompson series of class 5a for Monster.
%C G.f.: A(q) satisfies functional equation P(A(q)) = j(q^5), where P(x) = (x-1)^3 * (x^2 + 3*x + 36) and j is Klein's modular function. - _Michael Somos_, Jan 23 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A007253/b007253.txt">Table of n, a(n) for n = -1..1000</a> (terms -1..100 from G. A. Edgar)
%H J. H. Conway and S. P. Norton, <a href="https://doi.org/10.1112/blms/11.3.308">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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F G.f.: T5a(q) satisfies functional equation P5(T5a(q)) = j(q^5) - 744, where we used modular function j(q) from A000521 and polynomial P5(t) = t^5+30*t^3-100*t^2+105*t-780. _G. A. Edgar_, Mar 10 2017
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jan 23 2023
%e T5a = 1/q - 6*q + 20*q^2 + 15*q^3 + 36*q^4 - 84*q^6 + 195*q^7 + 100*q^8 + ...
%p with(numtheory): TOP := 23;
%p Order:=101;
%p g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));
%p g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));
%p delta := series(g2^3 - 27*g3^2, q=0, TOP);
%p j := series(1728 * g2^3 / delta, q=0, TOP);
%p # computation above of j is from A000521
%p P5 := t^5 + 30*t^3 - 100*t^2 + 105*t - 780;
%p subs(t = q^(-1) + x, P5) - subs(q=q^5, j - 744);
%p solve(%, x);
%p T5a := series(q^(-1)+%, q=0) assuming q > 0;
%p # _G. A. Edgar_, Mar 10 2017
%t eta[q_]:= q^(1/24)*QPochhammer[q]; e5B:= (eta[q]/eta[q^5])^6; e25a:= (eta[q]/eta[q^25]); a[n_]:= SeriesCoefficient[(1 + 5/e25a)*(1 + e5B) + 5*(e25a - 5/e25a)*(e5B/(e25a)^3), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* _G. C. Greubel_, Jan 25 2018 *)
%o (PARI) q='q+O('q^30); F=(1 + 5*q*eta(q^25)/eta(q))*(1 + (eta(q)/eta(q^5) )^6/q) + 5*(eta(q)/(q*eta(q^25)) - 5*q*eta(q^25)/eta(q))*(q^2* eta(q^25)^3 *eta(q)^3/eta(q^5)^6); Vec(F) \\ _G. C. Greubel_, Jun 12 2018
%Y Cf. A000521.
%K sign
%O -1,3
%A _N. J. A. Sloane_
%E More terms from _G. A. Edgar_, Mar 10 2017
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