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Expansion of (1+2*x+x^2)/(1-58*x+x^2).
1

%I #19 Apr 17 2016 03:24:28

%S 1,60,3480,201780,11699760,678384300,39334589640,2280727814820,

%T 132242878669920,7667806235040540,444600518753681400,

%U 25779162281478480660,1494746811806998196880,86669535922524416938380,5025338336694609184229160,291382953992364808268352900

%N Expansion of (1+2*x+x^2)/(1-58*x+x^2).

%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

%H Harvey P. Dale, <a href="/A004297/b004297.txt">Table of n, a(n) for n = 0..500</a>

%H J. M. Alonso, <a href="http://dx.doi.org/10.1007/978-1-4612-3142-4_1">Growth functions of amalgams</a>, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (58, -1).

%F a(0)=1, a(1)=60, a(2)=3480, a(n) = 58*a(n-1)-a(n-2). - _Harvey P. Dale_, Dec 30 2011

%F From _Colin Barker_, Apr 16 2016: (Start)

%F a(n) = sqrt(15/14)*((29+2*sqrt(210))^(-n)*(-1+(29+2*sqrt(210))^(2*n))) for n>0.

%F a(n) = 58*a(n-1) - a(n-2) for n>2.

%F (End)

%F a(n) = -(-1)^(2^n)/2 + sqrt(30/7)*sinh(n*log(29+2*sqrt(210))) + 1/2. - _Ilya Gutkovskiy_, Apr 16 2016

%t CoefficientList[Series[(1+2x+x^2)/(1-58x+x^2),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{58,-1},{60,3480},30]] (* _Harvey P. Dale_, Dec 30 2011 *)

%o (PARI) Vec((1+2*x+x^2)/(1-58*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_