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A007242 McKay-Thompson series of class 2a for the Monster group.

%I A007242 M5455 #90 Feb 16 2025 08:32:31
%S A007242 1,-492,-22590,-367400,-3764865,-28951452,-182474434,-990473160,
%T A007242 -4780921725,-20974230680,-84963769662,-321583404672,-1147744866180,
%U A007242 -3890805976500,-12601590210180,-39183052547592,-117437602167291,-340431109329600,-957251463332600,-2617490612355240,-6975126788952456,-18149106017123576,-46187557595906250
%N A007242 McKay-Thompson series of class 2a for the Monster group.
%C A007242 A more correct name would be: Expansion of replicable function of class 2a. See Alexander et al., 1992. - _N. J. A. Sloane_, Jun 12 2015
%C A007242 From "More on Replicable Functions": 'The fifth row consists of the class names. As stated above, the numbers are the replication orders. For those functions arising in Monstrous Moonshine, the letter corresponds to the relevant conjugacy class in the Monster in Atlas notation (or, if there is more than one class, the one with the first letter). For non-monstrous functions, the class names use lower case letters and, in accordance with Atlas notation, are arranged generally in descending order of Frobenian.'
%C A007242 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A007242 T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 425.
%D A007242 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007242 Seiichi Manyama, <a href="/A007242/b007242.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..500 from G. A. Edgar)
%H A007242 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 A007242 D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="/A007242/a007242_1.pdf">Completely replicable functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
%H A007242 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 A007242 Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%H A007242 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 A007242 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A007242 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A007242 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A007242 Sqrt(j-1728), where j is the j-function, see A000521.
%F A007242 A014708(2*n - 1) == a(n) (mod 256). That is, the coefficients of (T1A - T2a) are all divisible by 256. - _Michael Somos_, Jun 29 2011
%F A007242 Expansion of (-phi(-q)^12 - 30 * phi(-q)^8 * phi(q)^4 + 96 * phi(-q)^4 * phi(q)^8 - 64 * phi(q)^12) / f(-q)^12 where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Mar 17 2013
%F A007242 Expansion of (-8*(2*theta_2(0, q)^12-3*theta_2(0, q)^8*theta_3(0, q)^4-3*theta_3(0, q)^8*theta_2(0, q)^4+2*theta_3(0, q)^12))/(theta_3(0, q)^4*(theta_2(0, q)^4-theta_3(0, q)^4)*theta_2(0, q)^4) in powers of q.  Shows an analytic choice of the square root for complex q, 0 < |q| < 1. - _G. A. Edgar_, Mar 10 2017
%F A007242 G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/2). - _Seiichi Manyama_, Jul 02 2017
%F A007242 a(n) ~ -exp(2*Pi*sqrt(2*n)) / (2^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 09 2017
%e A007242 T2a = 1/q - 492*q - 22590*q^3 - 367400*q^5 - 3764865*q^7 - ...
%e A007242 196884 - (-492) = 197376 = 256 * 771, 21493760 - 0 = 256 * 83960, ...
%t A007242 a[ n_] :=  If[ n < 1, Boole[n == 0], SeriesCoefficient[ Sqrt[ 1728 (KleinInvariantJ[ Log[x] /(Pi I)] - 1) + O[x]^(2 n)], {x, 0, 2 n - 1}]] (* _Michael Somos_, Jun 29 2011 *)
%t A007242 nmax = 30; CoefficientList[Series[x^(1/2)*(-8*(2*EllipticTheta[2, 0, Sqrt[x]]^12 - 3*EllipticTheta[2, 0, Sqrt[x]]^8* EllipticTheta[3, 0, Sqrt[x]]^4 - 3*EllipticTheta[3, 0, Sqrt[x]]^8* EllipticTheta[2, 0, Sqrt[x]]^4 + 2*EllipticTheta[3, 0, Sqrt[x]]^12))/(EllipticTheta[3, 0, Sqrt[x]]^4*(EllipticTheta[2, 0, Sqrt[x]]^4 - EllipticTheta[3, 0, Sqrt[x]]^4)* EllipticTheta[2, 0, Sqrt[x]]^4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017, check of formula by G. A. Edgar *)
%t A007242 eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 55; f1A := (eta[q]/eta[q^2] )^24*(1 +256*(eta[q^2]/eta[q])^24)^3; A007242:= CoefficientList[ Series[(q*f1A - 1728*q + O[q]^nmax)^(1/2), {q, 0, 50}], q]; Table[ A007242[[n]], {n, 1, 50}] (* _G. C. Greubel_, May 09 2018 *)
%o A007242 (PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( ellj( x^2 * (1 + x * O(x^(2*n)) ) ) - 1728), 2*n - 1))} /* _Michael Somos_, Jun 29 2011 */
%o A007242 (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, n, -504 * sigma(k, 5) * x^k, 1 + x * O(x^n)) / eta(x + x * O(x^n))^12, n))} /* _Michael Somos_, Mar 17 2013 */
%Y A007242 (q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=12), A289063 (k=24).
%Y A007242 Cf. A000521, A014708, A289061.
%K A007242 sign
%O A007242 0,2
%A A007242 _N. J. A. Sloane_