A210873 - OEIS

%I #6 Oct 02 2013 16:26:12

%S 1,1,2,1,1,3,1,1,3,4,1,1,2,8,5,1,1,2,6,17,6,1,1,2,5,18,31,7,1,1,2,5,

%T 14,47,51,8,1,1,2,5,13,41,107,78,9,1,1,2,5,13,35,115,218,113,10,1,1,2,

%U 5,13,34,98,296,407,157,11,1,1,2,5,13,34,90,276,695,709,211,12

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

%C Column 1: 1,1,1,1,1,1,1,1,1...

%C Row sums: A083318 (1+2^n)

%C Alternating row sums:  A137470

%C Limiting row:  1,1,2,5,13,34,..., odd-indexed Fibonacci numbers

%C If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.

%C For a discussion and guide to related arrays, see A208510.

%F For a discussion and guide to related arrays, see A208510.

%F u(n,x)=x*u(n-1,x)+v(n-1,x)-1,

%F v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First six rows:

%e 1

%e 1...2

%e 1...1...3

%e 1...1...3....4

%e 1...1...2....8...5

%e 1...1...2....6...17...6

%e First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

%t u[1, x_] := 1; v[1, x_] := 1; z = 14;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;

%t v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A210872 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A210873 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

%Y Cf. A210872, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 29 2012