login
Expansion of 1 / ((1-x-x^2-x^3)*(1-x-x^3)).
1

%I #11 May 03 2017 08:40:06

%S 1,2,4,9,18,35,68,130,246,463,867,1617,3007,5579,10332,19107,35295,

%T 65140,120137,221444,407999,751453,1383641,2547116,4688106,8627504,

%U 15875390,29209560,53739655,98864470,181872110,334561861,615423932

%N Expansion of 1 / ((1-x-x^2-x^3)*(1-x-x^3)).

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.

%H G. C. Greubel, <a href="/A103321/b103321.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, 1, -2, -1, -1).

%F a(n) = A000073(n+4) - A000930(n+2).

%F a(n) = Sum_{k=0..n} A000073(k+2)*A000930(n-k).

%F a(0)=1, a(1)=2, a(2)=4, a(3)=9, a(4)=18, a(5)=35, a(n)=2*a(n-1)+a(n-3)- 2*a(n-4)-a(n-5)-a(n-6). - _Harvey P. Dale_, Nov 06 2011

%t CoefficientList[Series[1/((1-x-x^2-x^3)(1-x-x^3)),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,1,-2,-1,-1},{1,2,4,9,18,35},40] (* _Harvey P. Dale_, Nov 06 2011 *)

%o (PARI) x='x+O('x^50); Vec(1/((1-x-x^2-x^3)*(1-x-x^3))) \\ _G. C. Greubel_, May 02 2017

%K nonn

%O 0,2

%A _Ralf Stephan_, Feb 02 2005