A128517 - OEIS

%I #17 Feb 16 2025 08:33:04

%S 1,3,6,13,24,42,73,120,192,299,456,684,1007,1464,2100,2976,4176,5802,

%T 7993,10920,14808,19946,26688,35496,46944,61752,80826,105286,136536,

%U 176304,226725,290448,370704,471467,597600,755028,950980,1194216

%N McKay-Thompson series of class 18E for the Monster group with a(0) = 3.

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

%H G. C. Greubel, <a href="/A128517/b128517.txt">Table of n, a(n) for n = -1..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

%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>

%F Expansion of q^(-1) * (chi(-q^9) / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)))^3 in powers of q.

%F Euler transform of period 18 sequence [ 3, 0, 3, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 3, 0, 3, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - u*w * (6*(1 + v^2) - 10*v).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - u - v)^3 - u*v* (u + v - 1)^3.

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

%F G.f.: (1/x) * (Product_{k>0} P(x^k))^-3 where P(x) = (x^2 - x + 1) * (x^6 - x^3 + 1).

%F a(n) = A058535(n) unless n = 0.

%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%e G.f. = 1/q + 3 + 6*q + 13*q^2 + 24*q^3 + 42*q^4 + 73*q^5 + 120*q^6 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^9, q^9])^3, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]))^3, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(9*k)))^3,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)))^3, n))};

%Y Cf. A058535.

%K nonn,changed

%O -1,2

%A _Michael Somos_, Mar 06 2007