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A129507
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G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))*2.
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1
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0, 0, 0, 4, -4, 0, 8, -16, 12, 20, -80, 116, -4, -376, 884, -764, -1308, 5816, -9128, 648, 32008, -79608, 73100, 122020, -574044, 937456, -97748, -3366892, 8659492, -8276056, -13301640, 65261640, -109501476, 14857740, 395316672, -1042421076, 1028105156, 1582877768, -8052459580
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OFFSET
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0,4
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COMMENTS
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Expansion related to the asymptotic mean of the mean square error of a wireless channel.
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
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LINKS
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FORMULA
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G.f.: (sqrt(x(1+sqrt(x))^2+1)-sqrt(x(1-sqrt(x))^2+1))^2.
G.f.: 2+2x+2x^2-2*sqrt(1+2x+3x^2-2x^3+x^4).
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
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
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
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
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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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