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%I #34 Mar 12 2021 22:24:43
%S 1,-1,0,-1,1,-1,1,0,1,-2,1,-1,2,-2,3,-3,3,-4,4,-4,5,-4,4,-6,6,-7,7,-8,
%T 11,-11,10,-12,14,-15,15,-14,17,-20,19,-21,24,-26,30,-31,32,-37,38,
%U -40,45,-44,47,-54,56,-60,64,-68,79,-83,83,-92,100,-105,110,-112,123,-136,138,-147,160,-170,185,-194,203
%N Expansion of chi(-q) / chi(-q^7) 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).
%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
%C In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - _Vaclav Kotesovec_, Aug 31 2015
%H Seiichi Manyama, <a href="/A113297/b113297.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H F. G. Garvan and H. Yesilyurt, <a href="https://arxiv.org/abs/math/0605317">Shifted and shiftless partition identities II</a>, arXiv:math/0605317 [math.NT], 2003.
%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, p. 14.
%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^(-1/4) * eta(q) * eta(q^14) / ( eta(q^2) * eta(q^7) ) in powers of q.
%F Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, ...].
%F G.f. A(x) = G(x^7) * H(x^2) - x * G(x^2) * H(x^7) where G(x) and H(x) are the Rogers-Ramanujan functions.
%F G.f.: Product_{k>0} (1 + x^(7*k)) / (1 + x^k).
%F Expansion of chi(-q) / chi(-q^7) in powers of q where chi() is a Ramanujan theta function.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (224 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: Product_{k>0} P14(x^k) where P14 is the 14th cyclotomic polynomial.
%F Convolution inverse is A097793.
%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/7)) / (2^(3/2) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 31 2015
%e G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 + x^8 - 2*x^9 + x^10 - x^11 + ...
%e G.f. = q - q^5 - q^13 + q^17 - q^21 + q^25 + q^33 - 2*q^37 + q^41 + ...
%p seq(coeff(series(mul((1+x^(7*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..80); # _Muniru A Asiru_, Jul 29 2018
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^14] / (QPochhammer[ x^2] QPochhammer[ x^7]), {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^14 + A) / (eta(x^2 + A) * eta(x^7 + A)), n))};
%Y Cf. A097793.
%Y Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A261735 (m=8), A261733 (m=9), A145707 (m=10).
%K sign
%O 0,10
%A _Michael Somos_, Oct 23 2005