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%I #20 Mar 12 2021 22:24:44
%S 1,-1,1,1,-2,0,2,-2,-1,4,-1,-4,4,1,-6,3,6,-7,-3,10,-4,-10,12,6,-18,5,
%T 18,-20,-8,30,-10,-29,31,12,-46,17,44,-47,-20,68,-23,-66,72,31,-104,
%U 33,98,-107,-44,156,-51,-144,154,61,-220,75,206,-220,-90,310,-104,-290,312,131,-442,143,408,-437,-178,618,-202
%N Expansion of a parametrization of Ramanujan's continued fraction.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53
%H Seiichi Manyama, <a href="/A124242/b124242.txt">Table of n, a(n) for n = 0..10000</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 Euler transform of period 10 sequence [ -1, 1, 2, -1, -2, -1, 2, 1, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (2-u) * (2 - (2-u) * (2-v)).
%F Given g.f. A(x) =: k, then B(x) = (1-k) * (k / (2-k))^2, B(x^2) = (1-k)^2 * ((2-k) / k) where B(x) is the g.f. for A078905.
%F Expansion of f(-x^5, -x^10)^3 / (f(x, x^4) * f(-x^3, -x^7)^2) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Jan 06 2016
%F G.f.: Product_{k>0} ((1 - x^(10k-5)) / ((1 - x^(10k-3)) * (1 - x^(10k-7))))^2 * (1 - x^(10k-1)) * (1 - x^(10k-4)) * (1 - x^(10k-6)) * (1 - x^(10k-9) / ((1-x^(10k-2)) * (1-x^(10k-8))).
%F -a(n) = A112274(n) unless n = 0.
%F G.f.: 1 - r(q) * r(q^2)^2 where r() is the Rogers-Ramanujan continued fraction. - _Seiichi Manyama_, Apr 18 2017
%e G.f. = 1 - x + x^2 + x^3 - 2*x^4 + 2*x^6 - 2*x^7 - x^8 + 4*x^9 - x^10 - 4*x^11 + ...
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 1, -1, -2, 1, 2, 1, -2, -1, 1, 0}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Jan 06 2016 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + A)^[0, 1, -1, -2, 1, 2, 1, -2, -1, 1][k%10+1]), n))};
%Y Cf. A078905, A112274, A112803, A285348.
%K sign
%O 0,5
%A _Michael Somos_, Oct 27 2006
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