A058543 - OEIS

%I #26 Feb 16 2025 08:32:43

%S 1,-2,1,-4,8,-6,10,-16,18,-26,33,-40,58,-74,82,-112,147,-166,212,-268,

%T 316,-392,476,-560,695,-838,967,-1184,1430,-1648,1970,-2352,2731,

%U -3236,3803,-4404,5206,-6080,6984,-8192,9553,-10942,12709,-14736,16886,-19506,22448,-25648,29552

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

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

%H Vaclav Kotesovec, <a href="/A058543/b058543.txt">Table of n, a(n) for n = 0..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="https://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 chi(-x)^2 * chi(-x^3)^2 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Aug 18 2007

%F Expansion of q^(-1/3) * (eta(q) * eta(q^3) / (eta(q^2) * eta(q^6)))^2 in powers of q. - _Michael Somos_, Aug 18 2007

%F Euler transform of period 6 sequence [ -2, 0, -4, 0, -2, 0, ...]. - _Michael Somos_, Aug 18 2007

%F Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - u^2 * v - 4 * u. - _Michael Somos_, Aug 18 2007

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 4 / f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 18 2007

%F a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%e G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 10*x^6 - 16*x^7 + 18*x^8 - ...

%e T18e = 1/q - 2*q^2 + q^5 - 4*q^8 + 8*q^11 - 6*q^14 + 10*q^17 - 16*q^20 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^3, q^6])^2, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^2, n))}; /* _Michael Somos_, Aug 18 2007 */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%K sign,changed

%O 0,2

%A _N. J. A. Sloane_, Nov 27 2000