A129509 G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))/2.

0, 0, 0, 1, -1, 0, 2, -4, 3, 5, -20, 29, -1, -94, 221, -191, -327, 1454, -2282, 162, 8002, -19902, 18275, 30505, -143511, 234364, -24437, -841723, 2164873, -2069014, -3325410, 16315410, -27375369, 3714435, 98829168, -260605269, 257026289, 395719442, -2013114895, 3450787313, -572442080, -12414009687, 33423611731, -33865948418, -49805740764, 262037063892

Offset

0,7

Comments

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)/4. - Michael Somos, Mar 18 2014

Links

Table of n, a(n) for n=0..45.

A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications, Foundations and Trends in Communications and Information Theory, 1 (2004), 1-182.

Formula

G.f.: (sqrt(x(1+sqrt(x))^2+1)-sqrt(x(1-sqrt(x))^2+1))^2/4.

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

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]

a(n) = A129507(n)/4.

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

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

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

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

Conjecture: g.f. A(x) = x^3 * exp(Sum_{n >= 1} g(n, x)*(-x)^n/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*x^k. Cf. A167638. - Peter Bala, Sep 10 2024

Example

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 + ...

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A129507.

Sequence in context: A343312 A256640 A239466 * A375907 A375365 A341844

Adjacent sequences: A129506 A129507 A129508 * A129510 A129511 A129512

Keyword

sign

Author

Paul Barry, Apr 18 2007

Extensions

Prepended a(0)=a(1)=a(2)=0, Joerg Arndt, Jul 17 2013

Status

approved