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%I #14 Mar 30 2014 18:24:46
%S 0,0,0,1,-1,0,2,-4,3,5,-20,29,-1,-94,221,-191,-327,1454,-2282,162,
%T 8002,-19902,18275,30505,-143511,234364,-24437,-841723,2164873,
%U -2069014,-3325410,16315410,-27375369,3714435,98829168,-260605269,257026289,395719442,-2013114895,3450787313,-572442080,-12414009687,33423611731,-33865948418,-49805740764,262037063892
%N G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))/2.
%C Expansion related to the asymptotic mean of the mean square error of a wireless channel.
%C On page 11 of Tulino and Verdu is equation (1.17) F(x,z) = (sqrt(x(1+sqrt(z))^2+1) - sqrt(x(1-sqrt(z))^2+1))^2. G.f. is F(x,x)/4. - _Michael Somos_, Mar 18 2014
%H A. M. Tulino and S. Verdu, <a href="http://dx.doi.org/10.1561/0100000001">Random Matrix Theory and Wireless Communications</a>, Foundations and Trends in Communications and Information Theory, 1 (2004), 1-182.
%F G.f.: (sqrt(x(1+sqrt(x))^2+1)-sqrt(x(1-sqrt(x))^2+1))^2/4.
%F n*a(n) +(2*n-3)*a(n-1) +3*(n-3)*a(n-2) + (9-2*n)*a(n-3) +(n-6)*a(n-4)=0 if n>5. - _R. J. Mathar_, Nov 14 2011
%F Conjecture: g.f.: q^2*(1 - 1/G(0)) where G(k) = 1 + q/(1 + q^2 / G(k+1) ). [_Joerg Arndt_, Jul 17 2013]
%F a(n) = A129507(n)/4.
%F G.f.: 1 + x - (1 + x / (1 + x^2 / (1 + x / (1 + x^2 / ...)))). (continued fraction convergence is three power series terms per iteration) - _Michael Somos_, Mar 19 2014
%F G.f.: x * (1 - 1 / (1 - x + x^2 + x / (1 - x + x^2 + x / ...))). (continued fraction convergence is one power series term per iteration) - _Michael Somos_, Mar 18 2014
%F G.f.: x^2 * (1 - 1 / (1 + x - x^2 * (1 - 1 / (1 + x - x^2 * (1 - 1 / ...))))). (continued fraction convergence is two power series terms per iteration) - _Michael Somos_, Mar 30 2014
%F 0 = a(n)*(a(n+1) -5*a(n+2) +12*a(n+3) +11*a(n+4) +7*a(n+5)) + a(n+1)*(a(n+1) -2*a(n+2) -22*a(n+3) -21*a(n+4) -11*a(n+5)) + a(n+2)*(3*a(n+2) +17*a(n+3) +22*a(n+4) +12*a(n+5)) + a(n+3)*(-3*a(n+3) -2*a(n+4) +5*a(n+5)) + a(n+4)*(-a(n+4) +a(n+5)) if n>1. - _Michael Somos_, Mar 18 2014
%e G.f. = x^3 - x^4 + 2*x^6 - 4*x^7 + 3*x^8 + 5*x^9 - 20*x^10 + 29*x^11 - x^12 + ...
%t a[ n_] := SeriesCoefficient[ (1 + x + x^2 - Sqrt[1 + 2*x + 3*x^2 - 2*x^3 + x^4]) / 2, {x, 0, n}]; (* _Michael Somos_, Mar 18 2014 *)
%o (PARI) x='x+O('x^66); Vec( (1+x+x^2-sqrt(1+2*x+3*x^2-2*x^3+x^4))/2 ) \\ _Joerg Arndt_, Jul 17 2013
%Y Cf. A129507.
%K sign
%O 0,7
%A _Paul Barry_, Apr 18 2007
%E Prepended a(0)=a(1)=a(2)=0, _Joerg Arndt_, Jul 17 2013
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