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A029841 McKay-Thompson series of class 8E for the Monster group.

%I A029841 #47 Sep 02 2025 06:04:38
%S A029841 1,4,2,-8,-1,20,-2,-40,3,72,2,-128,-4,220,-4,-360,5,576,8,-904,-8,
%T A029841 1384,-10,-2088,11,3108,12,-4552,-15,6592,-18,-9448,22,13392,26,
%U A029841 -18816,-29,26216,-34,-36224,38,49700,42,-67728,-51,91688
%N A029841 McKay-Thompson series of class 8E for the Monster group.
%C A029841 A Hauptmodul for Gamma'_0(8).
%C A029841 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A029841 A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
%H A029841 Seiichi Manyama, <a href="/A029841/b029841.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A029841 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. See page 336.
%H A029841 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 A029841 J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%H A029841 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 A029841 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A029841 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A029841 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A029841 G.f.: ( Product_{k>0} (1 + q^(2*k - 1)) / (1 + q^(2*k)) )^4.
%F A029841 Expansion of q^(1/4) * (1 + k) / k^(1/2) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus. - _Michael Somos_, Aug 01 2011
%F A029841 Expansion of q^(1/2) * 4 / k in powers of q where q is Jacobi's nome and k is the elliptic modulus. - _Michael Somos_, Aug 01 2011 and Feb 28 2012
%F A029841 Expansion of (phi(x) / psi(x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%F A029841 Expansion of q^(1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)^2))^4 in powers of q. - _Michael Somos_, Aug 01 2011
%F A029841 Euler transform of period 4 sequence [4, -8, 4, 0, ...]. - _Michael Somos_, Mar 18 2004
%F A029841 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 16 + 8*v + v^2 - u^2*v. - _Michael Somos_, Mar 18 2004
%F A029841 G.f. A(q) satisfies A(q) = sqrt(A(q^2))+4*q/sqrt(A(q^2)). - _Joerg Arndt_, Aug 06 2011
%F A029841 A112143(n) = (-1)^n * a(n). a(2*n) = A029839(n). a(2*n + 1) = 4 * A079006(n). - _Michael Somos_, Mar 27 2004.
%F A029841 Convolution inverse of A001938. Convolution square of A029839. Convolution square is A029845.
%e A029841 G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - x^4 + 20*x^5 - 2*x^6 - 40*x^7 + 3*x^8 + ...
%e A029841 T8E = 1/q + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*x^15 + ...
%t A029841 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 / Sqrt[m], {q, 0, n - 1/2}]]; (* _Michael Somos_, Aug 01 2011 *)
%t A029841 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 2 (1 + Sqrt[m]) / m^(1/4), {q, 0, n/2 - 1/4}]]; (* _Michael Somos_, Aug 01 2011 *)
%t A029841 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 / (QPochhammer[ x] QPochhammer[x^4]^2))^4, {x, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%o A029841 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A))^2)^4, n))};
%o A029841 (PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = (4*x + A) / sqrt(A)); polcoeff(A, n))};
%Y A029841 Cf. A001938, A029839, A029845, A079006, A112143.
%K A029841 sign,easy,nice,changed
%O A029841 0,2
%A A029841 _N. J. A. Sloane_