%I #18 Mar 12 2021 22:24:44
%S 1,-1,0,-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 with a(0) = -1.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A132322/b132322.txt">Table of n, a(n) for n = -1..1000</a>
%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 q^-1 * chi(-q) * chi(-q^23) in powers of q where chi() is a Ramanujan theta function.
%F Expansion of eta(q) * eta(q^23) / (eta(q^2) * eta(q^46)) in powers of q.
%F Euler transform of period 46 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 2 * u + 2 * u * v.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A092833.
%F G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(23*k)))^-1.
%F a(n) = A058688(n) unless n = 0.
%F Convolution inverse of A092833. - _Michael Somos_, Mar 23 2015
%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e G.f. = 1/q - 1 - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 - 2*q^10 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^23, q^46] / q, {q, 0, n}]; (* _Michael Somos_, Mar 23 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + A)), n))};
%Y Cf. A058688, A092833.
%K sign
%O -1,9
%A _Michael Somos_, Aug 18 2007