|
%I #43 Dec 25 2021 16:38:36
%S 1,-3,3,-2,3,-6,10,-12,15,-22,30,-36,44,-60,78,-96,117,-150,190,-228,
%T 276,-340,420,-504,603,-732,885,-1052,1245,-1488,1770,-2088,2454,
%U -2902,3420,-3996,4666,-5460,6378,-7400,8583,-9972,11566,-13344,15378,-17752,20448,-23472,26904,-30876
%N McKay-Thompson series of class 18C for the Monster group with a(0) = -3.
%C A058533, A123676, A215412, A058644, A215413 are all essentially the same sequence. - _N. J. A. Sloane_, Aug 09 2012
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A123676/b123676.txt">Table of n, a(n) for n = -1..1000</a>
%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). See Table 4 18C.
%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>
%F Expansion of -4 + psi(q) / (q * psi(q^9)) + 3 * q * psi(q^9) / psi(q) in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Aug 09 2012
%F Expansion of (1/q) * (chi(-q) * chi(-q^9))^3 / chi(-q^3)^2 in powers of q where chi() is a Ramanujan theta function.
%F Expansion of b(q) * c(q^3) / (b(q^2) * c(q^6)) in powers of q where b(), c() are cubic AGM theta functions.
%F Given g.f. A(x), then B(x) = 1/A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v - u * v * (6 + 4*v).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A123629.
%F a(n) = A058533(n) = A215412(n) = A215413(n) unless n=0. - _Michael Somos_, Aug 09 2012
%F Convolution inverse of A123629.
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 12 2015
%e 1/q - 3 + 3*q - 2*q^2 + 3*q^3 - 6*q^4 + 10*q^5 - 12*q^6 + 15*q^7 - 22*q^8 + ...
%t QP = QPochhammer; s = (QP[q]*(QP[q^9]/(QP[q^2]*QP[q^18])))^3*(QP[q^6]/ QP[q^3])^2 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015, adapted from PARI *)
%t nmax = 60; CoefficientList[Series[Product[(1+x^(3*k))^2 / ( (1+x^k)^3 * (1+x^(9*k))^3), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 12 2015 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x*O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A) / (eta(x^2 + A) * eta(x^18 + A)))^3 * (eta(x^6 + A) / eta(x^3 + A))^2, n))}
%Y Cf. A058533, A123629, A215412, A215413.
%K sign
%O -1,2
%A _Michael Somos_, Oct 05 2006
|
