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%I #32 Mar 12 2021 22:25:44
%S 1,-8,28,-64,142,-352,792,-1536,2917,-5744,10868,-19200,33414,-58816,
%T 101256,-167936,275314,-452392,732748,-1160064,1819808,-2851104,
%U 4421064,-6752256,10236407,-15476272,23215192,-34450944,50811638,-74701632,109138272,-158171136
%N Expansion of q * (chi(-q) / chi(-q^4))^8 in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A134747/b134747.txt">Table of n, a(n) for n = 1..1000</a>
%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 k * (1 - k) / ( 4 * (1 + k) ) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
%F Expansion of ( (eta(q) * eta(q^8)) / (eta(q^2) * eta(q^4)) )^8 in powers of q.
%F Euler transform of period 8 sequence [ -8, 0, -8, 8, -8, 0, -8, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * u*w * (v*w-1) * (v*u-1) - (v - u^2) * (v - w^2).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: x * ( Product_{k>0} (1 + x^(4*k)) / (1 + x^k) )^8.
%F Convolution inverse of A131123.
%F Convolution 8th power of A261734. - _Michael Somos_, Oct 16 2015
%F a(n) ~ -(-1)^n * exp(Pi*sqrt(2*n)) / (2^(21/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 10 2018
%F Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = -8 - 6*sqrt(2) + (3/2)*sqrt(60 + 43*sqrt(2)). - _Simon Plouffe_, Mar 04 2021
%e G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 142*q^5 - 352*q^6 + 792*q^7 - 1536*q^8 + ...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^2] QPochhammer[ -q^4, q^4])^8, {q, 0, n}]; (* _Michael Somos_, Oct 16 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( (eta(x + A) * eta(x^8 + A)) / (eta(x^2 + A) * eta(x^4 + A)) )^8, n))};
%Y Cf. A131123, A261734.
%K sign
%O 1,2
%A _Michael Somos_, Nov 07 2007
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