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A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

%I A210872 #6 Mar 31 2012 20:30:51
%S A210872 1,0,1,0,2,1,0,1,5,1,0,1,4,9,1,0,1,3,12,14,1,0,1,3,9,29,20,1,0,1,3,8,
%T A210872 27,60,27,1,0,1,3,8,22,74,111,35,1,0,1,3,8,21,63,181,189,44,1,0,1,3,8,
%U A210872 21,56,178,399,302,54,1,0,1,3,8,21,55,154,474,806,459,65,1,0,1
%N A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.
%C A210872 Column 1: 1,0,0,0,0,0,0,0,0,...
%C A210872 Row sums: A000225 (-1+2^n)
%C A210872 Alternating row sums:  (-1)*A077973
%C A210872 Limiting row:  0,1,3,8,21,..., even-indexed Fibonacci numbers
%C A210872 If the term in row n and column k is written as U(n,k), then U(n,n-1)=A000096 and U(n,n-2)=A086274.
%C A210872 For a discussion and guide to related arrays, see A208510.
%F A210872 u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
%F A210872 v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210872 where u(1,x)=1, v(1,x)=1.
%F A210872 Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.
%e A210872 First six rows:
%e A210872 1
%e A210872 0...1
%e A210872 0...2...1
%e A210872 0...1...5...1
%e A210872 0...1...4...9....1
%e A210872 0...1...3...12...14...1
%e A210872 First three polynomials u(n,x): 1, x, 2x + x^2.
%t A210872 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210872 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
%t A210872 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210872 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210872 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210872 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210872 TableForm[cu]
%t A210872 Flatten[%]    (* A210872 *)
%t A210872 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210872 TableForm[cv]
%t A210872 Flatten[%]    (* A210873 *)
%t A210872 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
%t A210872 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
%t A210872 Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
%t A210872 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)
%Y A210872 Cf. A210873, A208510.
%K A210872 nonn,tabl
%O A210872 1,5
%A A210872 _Clark Kimberling_, Mar 29 2012