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A152944 McKay-Thompson series of class 17A for the Monster group with a(0) = 2. 3

%I

%S 1,2,7,14,29,50,92,148,246,386,603,904,1367,1996,2914,4160,5924,8290,

%T 11581,15942,21878,29712,40184,53876,71979,95436,126097,165556,216594,

%U 281848,365548,471808,607050,777794,993528,1264338,1604434,2029026

%N McKay-Thompson series of class 17A for the Monster group with a(0) = 2.

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

%H Vaclav Kotesovec, <a href="/A152944/b152944.txt">Table of n, a(n) for n = -1..10000</a>

%H M. 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) * ((psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)))^2 in powers of q where phi(), psi(), f() are Ramanujan theta functions.

%F Expansion of q^(-1) * (F(q) - q^4 / F(q))^2 / (chi(-q) * chi(-q^17))^4 in powers of q where F(q) = G(q^17) / G(q), G(q) = chi(q) * chi(-q^2) and chi() is a Ramanujan theta function.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (17 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = A058530(n) unless n = 0. Convolution square of A058639.

%F a(n) ~ exp(4*Pi*sqrt(n/17)) / (sqrt(2) * 17^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e G.f. = 1/q + 2 + 7*q + 14*q^2 + 29*q^3 + 50*q^4 + 92*q^5 + 148*q^6 + 246*q^7 + ...

%t QP = QPochhammer; s = (QP[q^4]^2*(QP[q^34]^5/(QP[q]*QP[q^2]* QP[q^17]^3* QP[q^68]^2)) - q^4*QP[q^2]^5*(QP[q^68]^2/(QP[q]^3*QP[q^4]^2*QP[q^17]* QP[q^34])))^2 + O[q]^40; CoefficientList[s, q] (* _Jean-Fran├žois Alcover_, Nov 15 2015, adapted from PARI *)

%t nmax = 60; CoefficientList[Series[(Product[(1+x^k) * (1+x^(2*k))^2 * (1+x^(17*k)) * (1+x^(34*k-17))^2, {k, 1, nmax}] - x^4*Product[(1+x^k) * (1+x^(2*k-1))^2 * (1+x^(17*k)) * (1+x^(34*k))^2, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 01 2017 *)

%t a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^17] - EllipticTheta[ 2, 0, q^17] EllipticTheta[ 3, 0, q]) / (QPochhammer[ q] QPochhammer[ q^16]))^2 / (4 q^(3/2)), {q, 0, n}]; (* _Michael Somos_, Sep 06 2018 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^17 + A)^3 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^17 + A) * eta(x^34 + A)))^2, n))};

%Y Cf. A058530, A058639.

%K nonn

%O -1,2

%A _Michael Somos_, Dec 15 2008

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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)