login
G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))*2.
1

%I #26 Jan 30 2020 21:29:16

%S 0,0,0,4,-4,0,8,-16,12,20,-80,116,-4,-376,884,-764,-1308,5816,-9128,

%T 648,32008,-79608,73100,122020,-574044,937456,-97748,-3366892,8659492,

%U -8276056,-13301640,65261640,-109501476,14857740,395316672,-1042421076,1028105156,1582877768,-8052459580

%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). - _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.

%F G.f.: 2+2x+2x^2-2*sqrt(1+2x+3x^2-2x^3+x^4).

%F D-finite with recurrence: 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 05 2012

%F Conjecture: g.f.: 4*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) = A129509(n)*4.

%F G.f.: 4 * (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.: 4*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 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. = 4*x^3 - 4*x^4 + 8*x^6 - 16*x^7 + 12*x^8 + 20*x^9 - 80*x^10 + 116*x^11 + ...

%t a[ n_] := SeriesCoefficient[ 2 (1 + x + x^2 - Sqrt[1 + 2*x + 3*x^2 - 2*x^3 + x^4]), {x, 0, n}]; (* _Michael Somos_, Mar 18 2014 *)

%o (PARI) x='x+O('x^66); Vec( 2+2*x+2*x^2-2*sqrt(1+2*x+3*x^2-2*x^3+x^4) ) \\ _Joerg Arndt_, Jul 17 2013

%Y Cf. A129509.

%K sign

%O 0,4

%A _Paul Barry_, Apr 18 2007