|
%I #32 Mar 12 2021 22:24:42
%S 1,0,0,-1,1,-1,1,-1,2,-2,2,-2,3,-3,3,-4,5,-5,5,-6,7,-8,8,-10,12,-12,
%T 13,-15,17,-18,19,-22,25,-27,28,-32,36,-38,41,-46,51,-54,58,-64,71,
%U -76,81,-89,99,-105,112,-123,134,-143,153,-167,182,-194,207,-225,244,-260,277,-301,325,-346,369,-398,429,-458
%N McKay-Thompson series of class 46A for the Monster group.
%C Also McKay-Thompson series of class 46B for the Monster group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A058688/b058688.txt">Table of n, a(n) for n = -1..5000</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).
%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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of 1 + (1/q) * chi(-q) * chi(-q^23) in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Jun 07 2006
%F G.f.: 1 + (1/x) * Product_{k>0} 1 / ((1 + x^k) * (1 + x^(23*k))).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - v + 2 - 2*u + u^2 - v*u^2. - _Michael Somos_, Feb 14 2007
%F a(n) = A132322(n) unless n=0.
%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%F Expansion of 1 + eta(q)*eta(q^23)/(eta(q^2)*eta(q^46)) in powers of q. - _G. C. Greubel_, Jun 16 2018
%e T46A = 1/q - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 + ...
%t QP = QPochhammer; s = q + QP[q]*(QP[q^23]/(QP[q^2]*QP[q^46])) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 13 2015, adapted from PARI *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(1+eta[q] *eta[q^23]/(eta[q^2]*eta[q^46])), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 16 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( x + eta(x + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + A)), n))} /* _Michael Somos_, Feb 14 2007 */
%Y Cf. A000521, A007240, A007241, A007267, A014708, A045478.
%Y Cf. A132322.
%K sign
%O -1,9
%A _N. J. A. Sloane_, Nov 27 2000
|
