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A232772 Expansion of (psi(x)^2 / (phi(-x) * phi(x^2)))^2 in powers of x where phi(), psi() are Ramanujan theta functions. 3

%I

%S 1,8,30,80,197,472,1046,2160,4306,8360,15712,28656,51127,89552,153926,

%T 259904,432336,709728,1150142,1841200,2915546,4570904,7097622,

%U 10921184,16664073,25228176,37907758,56553936,83806768,123405752,180611558,262799248,380275604

%N Expansion of (psi(x)^2 / (phi(-x) * phi(x^2)))^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A232772/b232772.txt">Table of n, a(n) for n = 0..2500</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 M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">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^(-1/2) * (eta(q^2)^7 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.

%F Euler transform of period 8 sequence [ 8, -6, 8, 4, 8, -6, 8, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 g(t) where q = eqp(2 Pi i t) and g() is the g.f. of A233458.

%F a(n) = A215349(2*n + 1) = A215348(2*n + 1). 2 * a(n) = A212318(2*n + 1) = - A232358(2*n + 1).

%F a(n) ~ exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%e G.f. = 1 + 8*x + 30*x^2 + 80*x^3 + 197*x^4 + 472*x^5 + 1046*x^6 + 2160*x^7 + ...

%e G.f. = q + 8*q^3 + 30*q^5 + 80*q^7 + 197*q^9 + 472*q^11 + 1046*q^13 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^4 / (16 q^(1/2)) / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]

%t nmax=60; CoefficientList[Series[Product[((1-x^k)^3 * (1+x^k)^7 * (1+x^(4*k))^2 / (1-x^(4*k))^3)^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))}

%Y Cf. A212318, A215348, A215348, A232358, A233458.

%K nonn

%O 0,2

%A _Michael Somos_, Nov 30 2013

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)