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A215348 Expansion of q * phi(q) * psi(q^8) / (phi(-q) * phi(q^4)) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A215348 #21 Feb 16 2025 08:33:18
%S A215348 1,4,8,16,30,48,80,128,197,312,472,704,1046,1504,2160,3072,4306,6036,
%T A215348 8360,11488,15712,21264,28656,38400,51127,67864,89552,117632,153926,
%U A215348 200352,259904,335872,432336,554952,709728,904784,1150142,1457136,1841200,2320128
%N A215348 Expansion of q * phi(q) * psi(q^8) / (phi(-q) * phi(q^4)) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A215348 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A215348 G. C. Greubel, <a href="/A215348/b215348.txt">Table of n, a(n) for n = 1..1000</a>
%H A215348 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], 2015-2016.
%H A215348 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215348 Expansion of q * (f(q) * f(-q^16) / (f(-q) * f(q^4)))^2 = q * (chi(-q^2) * chi(-q^4) / (chi(-q) * chi(-q^8))^2)^2 in powers of q where chi(), f() are Ramanujan theta functions.
%F A215348 Expansion of (eta(q^2)^3 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^3))^2 in powers of q.
%F A215348 Euler transform of period 16 sequence [ 4, -2, 4, -2, 4, -2, 4, 4, 4, -2, 4, -2, 4, -2, 4, 0, ...].
%F A215348 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A215346.
%F A215348 a(n) = -(-1)^n * A215349(n). a(2*n) = 4 * A107035(n). Convolution inverse of A215346.
%F A215348 a(n) ~ exp(sqrt(n)*Pi) / (8*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A215348 q + 4*q^2 + 8*q^3 + 16*q^4 + 30*q^5 + 48*q^6 + 80*q^7 + 128*q^8 + 197*q^9 + ...
%t A215348 nmax=60; CoefficientList[Series[Product[((1+x^k)^3 * (1-x^k) * (1+x^(8*k))^2 / (1-x^(8*k)))^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%t A215348 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]*EllipticTheta[2, 0, q^4]/(2*EllipticTheta[3, 0, -q]*EllipticTheta[3, 0, q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 07 2017 *)
%o A215348 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^3))^2, n))}
%Y A215348 Cf. A107035, A215346, A215349.
%K A215348 nonn
%O A215348 1,2
%A A215348 _Michael Somos_, Aug 08 2012