cp's OEIS Frontend

A029839 McKay-Thompson series of class 16B for the Monster group.

%I A029839 #63 Feb 16 2025 08:32:35
%S A029839 1,2,-1,-2,3,2,-4,-4,5,8,-8,-10,11,12,-15,-18,22,26,-29,-34,38,42,-51,
%T A029839 -56,66,78,-85,-98,109,120,-139,-156,176,202,-222,-250,279,306,-346,
%U A029839 -384,429,482,-530,-590,650,714,-797,-876,972,1080,-1180,-1304,1431,1562,-1728,-1892,2078,2290,-2496
%N A029839 McKay-Thompson series of class 16B for the Monster group.
%C A029839 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A029839 In [Klein and Fricke 1890], the g.f. A(q)/2 is denoted by mu. On page 613 special values given are mu(i infinity) = infinity, mu(0) = 1, mu(2) = -1 and on page 615 properties given are mu(omega+1) = -i mu(omega), mu(-1/omega) = (mu(omega)+1)/(mu(omega)-1). - _Michael Somos_, Nov 09 2014
%H A029839 Seiichi Manyama, <a href="/A029839/b029839.txt">Table of n, a(n) for n = 0..10000</a>
%H A029839 R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
%H A029839 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 A029839 F. Klein and R. Fricke, <a href="https://archive.org/stream/vorlesungenberd03kleigoog#page/n637/mode/2up">Vorlesungen über die theorie der elliptischen modulfunctionen</a>, Teubner, Leipzig, 1890, Vol. 1, see pp. 613, 615, 675.
%H A029839 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 A029839 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A029839 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A029839 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A029839 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A029839 Expansion of q times normalized Hauptmodul for Gamma(4) in powers of q^4.
%F A029839 Expansion of q^(1/4) * eta(q^2)^6 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F A029839 Euler transform of period 4 sequence [2, -4, 2, 0, ...].
%F A029839 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x / A(x^2). - _Michael Somos_, Mar 08 2004
%F A029839 G.f.: Product_{k>0} ((1 + x^(2*k-1)) / (1 + x^(2*k)))^2.
%F A029839 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^2 - u^2*v. - _Michael Somos_, May 14 2004
%F A029839 Given g.f. A(x), then B(q) = A(q^4) / (2*q) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - _Michael Somos_, Oct 04 2006
%F A029839 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u6 + u2)^2 - u1*u2*u3*u6. - _Michael Somos_, Oct 04 2006
%F A029839 Convolution inverse of A079006.
%F A029839 Expansion of q^(1/4) * 2 / k(q)^(1/2) in powers of Jacobi nome q where k() is the elliptic modulus.
%F A029839 Expansion of q^(1/2) * 2 * (1 + k'(q)) / k(q) in powers of q^2. - _Michael Somos_, Nov 09 2014
%F A029839 Expansion of phi(x) / psi(x^2) = phi(x)^2 / psi(x)^2 = psi(x)^2 / psi(x^2)^2 = phi(-x^2)^2 / psi(-x)^2 = chi(-x^2)^4 / chi(-x)^2 = chi(x)^2 * chi(-x^2)^2 = chi(x)^4 * chi(-x)^2 = f(x)^2 / f(-x^4)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A029839 Expansion of continued fraction 1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)) in powers of x^4. - _Michael Somos_, Apr 27 2008
%F A029839 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007096.
%F A029839 a(n) = (-1)^n * A082304(n). Convolution square is A029841. - _Michael Somos_, Jul 05 2014
%F A029839 From _Peter Bala_, Jan 09 2021: (Start)
%F A029839 A(q) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+1)) / Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)).
%F A029839 A(q) = ( 1 + q/(1 + (q + q^2)/(1 + q^3/(1 + (q^2 + q^4)/(1 + q^5/(1 + ... ))))) )^2. See Agarwal, p. 285.
%F A029839 A(q) = B(q)^2, where B(q) is the g.f. of A029838. (End)
%F A029839 abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Feb 07 2023
%e A029839 G.f. = 1 + 2*x - x^2 - 2*x^3 + 3*x^4 + 2*x^5 - 4*x^6 - 4*x^7 + 5*x^8 + 8*x^9 + ...
%e A029839 T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
%t A029839 a[0] = 1; a[n_] := Module[{A, m}, If[n < 0, 0, A = 1; m = 1; While[m <= n, m *= 2; A = A /. x -> x^2; A = Sqrt[A + 4*x/A]]; SeriesCoefficient[A, {x, 0, n}]]]; Table[a[n], {n, 0, 58}] (* _Jean-François Alcover_, Mar 12 2014, after PARI *)
%t A029839 a[ n_] := SeriesCoefficient[ 2 q^(1/4) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)
%t A029839 QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]^4) + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%o A029839 (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))^2, n))};
%o A029839 (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 = sqrt(A + 4*x/A)); polcoeff(A, n))};
%Y A029839 Cf. A079006, A082304, A029838.
%Y A029839 Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^b: this sequence (b=1), A029839 (b=2), A029840 (b=3), A029841 (b=4), A029842 (b=5), A029843 (b=6), A029844 (b=7).
%K A029839 sign,easy
%O A029839 0,2
%A A029839 _N. J. A. Sloane_
%E A029839 Additional comments from _Michael Somos_, Jul 11 2002