A062242 - OEIS

%I #45 Mar 12 2021 22:24:42

%S 1,1,1,-1,-1,0,1,2,0,-2,-3,-1,4,4,1,-4,-6,-1,5,8,1,-8,-10,-2,11,14,4,

%T -14,-19,-4,17,24,4,-23,-31,-6,31,40,9,-38,-50,-10,46,63,11,-60,-79,

%U -16,77,98,21,-92,-122,-24,112,150,28,-140,-183,-36,173,224,46,-208,-273,-54,249,329,62,-304,-396,-78,370

%N McKay-Thompson series of class 18D for the Monster group.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C Number 7 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014

%C There is a typo in the entry for this q-series in Table I of Yang 2004. The exponent of 18 should be 3. - _Michael Somos_, Jul 21 2014

%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(18). [Yang 2004] - _Michael Somos_, Jul 21 2014

%H Seiichi Manyama, <a href="/A062242/b062242.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%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 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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k-1)). - _Michael Somos_, Mar 17 2004

%F Expansion of chi(-q^3)^3 / chi(-q) in powers of q where chi() is a Ramanujan theta function.

%F Expansion of q^(1/3) * c(q) / c(q^2) in powers of q where c() is a cubic AGM theta function. - _Michael Somos_, Oct 17 2006

%F Expansion of q^(1/3) * eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q. - _Michael Somos_, Mar 05 2004

%F Euler transform of period 6 sequence [ 1, 0, -2, 0, 1, 0, ...]. - _Michael Somos_, Mar 05 2004

%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 2*u  +v^2 - u^2*v. - _Michael Somos_, Mar 17 2004

%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - v + v^2) * u^3 - (4 + 2*v + v^2) * v. - _Michael Somos_, Aug 11 2007

%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u^6 + v^6 - u^5*v^5 + 5*u^4*v^4 - 20*u^3*v^3 + 20*u^2*v^2 - 16*u*v + 5*u^2*v^5 + 5*u^5*v^2 - 10*u^4*v - 10*u*v^4. - _Michael Somos_, Aug 11 2007

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128128.

%F G.f.: 1 + x*(1+x)/(1 + x^2*(1+x^2)/(1 + x^3*(1+x^3)/(1 + x^4*(1+x^4)/(1 + x^5*(1+x^5)/(1 + ...))))), a continued fraction. - _Paul D. Hanna_, Jul 09 2013

%F a(n) = (-1)^n * A062244(n). a(2*n) = A132179(n). a(2*n + 1) = A092848(n).

%F Convolution inverse of A092848.

%e G.f. = 1 + x + x^2 - x^3 - x^4 + x^6 + 2*x^7 - 2*x^9 - 3*x^10 - x^11 + 4*x^12 + ...

%e T18D = 1/q + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 - x^32 +  ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* _Michael Somos_, Jul 21 2014 *)

%o (PARI) {a(n) = local(A,m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=3; A = subst(A, x, x^3) / x; A = (x*A * (4 + 2*A + A^2) / (1 - A + A^2))^(1/3)); polcoeff(A, n))};

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3), n))};

%o (PARI) /* Continued Fraction Expansion: */

%o {a(n)=local(CF); CF=1+x; for(k=0, n, CF=1 + x^(n-k+1)*(1 + x^(n-k+1))/(CF+x*O(x^n))); polcoeff(CF, n)} \\ _Paul D. Hanna_, Jul 09 2013

%Y Cf. A062244, A092848, A132179.

%K sign

%O 0,8

%A _N. J. A. Sloane_, Jun 30 2001