A128516 - OEIS

%I #17 Mar 12 2021 22:24:44

%S 1,4,10,24,51,100,190,340,585,984,1606,2564,4022,6188,9382,14044,

%T 20746,30308,43836,62784,89153,125588,175542,243656,335988,460388,

%U 627178,849676,1145024,1535416,2049200,2722544,3601681,4745208,6227276,8141656

%N McKay-Thompson series of class 14C for the Monster group with a(0) = 4.

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

%H Seiichi Manyama, <a href="/A128516/b128516.txt">Table of n, a(n) for n = -1..10000</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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-1) * (chi(-q^7) / chi(-q))^4 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)))^4 in powers of q.

%F Euler transform of period 14 sequence [ 4, 0, 4, 0, 4, 0, 0, 0, 4, 0, 4, 0, 4, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = f(t) where q = exp(2 Pi i t).

%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 * (8*(1 + v^2) - 16*v).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v + 1) * (u + v) * (u - v)^2 - u*v * (u - 1) * (v - 1) * (u*v - 8*(u+v) + 1).

%F G.f.: (1/x) * (Product_{k>0} P(x^k))^-4 where P(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 is the 14th cyclotomic polynomial.

%F a(n) = A058504(n) unless n = 0.

%F a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (2^(3/4) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015

%e G.f. = 1/q + 4 + 10*q + 24*q^2 + 51*q^3 + 100*q^4 + 190*q^5 + 340*q^6 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^7, q^7])^4, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^7] / (QPochhammer[ q] QPochhammer[ q^14]))^4, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(7*k)))^4,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)))^4, n))};

%Y Cf. A058504.

%K nonn

%O -1,2

%A _Michael Somos_, Mar 06 2007