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%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
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