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

%I A210873 #6 Oct 02 2013 16:26:12
%S A210873 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 A210873 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 A210873 5,13,34,98,296,407,157,11,1,1,2,5,13,34,90,276,695,709,211,12
%N A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.
%C A210873 Column 1: 1,1,1,1,1,1,1,1,1...
%C A210873 Row sums: A083318 (1+2^n)
%C A210873 Alternating row sums:  A137470
%C A210873 Limiting row:  1,1,2,5,13,34,..., odd-indexed Fibonacci numbers
%C A210873 If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.
%C A210873 For a discussion and guide to related arrays, see A208510.
%F A210873 For a discussion and guide to related arrays, see A208510.
%F A210873 u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
%F A210873 v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210873 where u(1,x)=1, v(1,x)=1.
%e A210873 First six rows:
%e A210873 1
%e A210873 1...2
%e A210873 1...1...3
%e A210873 1...1...3....4
%e A210873 1...1...2....8...5
%e A210873 1...1...2....6...17...6
%e A210873 First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
%t A210873 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210873 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
%t A210873 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210873 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210873 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210873 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210873 TableForm[cu]
%t A210873 Flatten[%]    (* A210872 *)
%t A210873 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210873 TableForm[cv]
%t A210873 Flatten[%]    (* A210873 *)
%t A210873 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
%t A210873 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
%t A210873 Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
%t A210873 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)
%Y A210873 Cf. A210872, A208510.
%K A210873 nonn,tabl
%O A210873 1,3
%A A210873 _Clark Kimberling_, Mar 29 2012