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A209689 Triangle of coefficients of polynomials u(n,x) jointly generated with A209690; see the Formula section.

%I A209689 #5 Mar 30 2012 18:58:15
%S A209689 1,0,2,0,2,3,0,1,6,4,0,1,4,13,5,0,1,3,13,24,6,0,1,3,9,35,40,7,0,1,3,8,
%T A209689 28,81,62,8,0,1,3,8,22,82,167,91,9,0,1,3,8,21,64,217,315,128,10,0,1,3,
%U A209689 8,21,56,188,519,554,174,11,0,1,3,8,21,55,155,529,1136,921
%N A209689 Triangle of coefficients of polynomials u(n,x) jointly generated with A209690; see the Formula section.
%C A209689 Combinatorial limit of rows:  even-indexed Fibonacci numbers. For a discussion and guide to related arrays, see A208510.
%F A209689 u(n,x)=x*u(n-1,x)+x*v(n-1,x),
%F A209689 v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A209689 where u(1,x)=1, v(1,x)=1.
%e A209689 First five rows:
%e A209689 1
%e A209689 0...2
%e A209689 0...2...3
%e A209689 0...1...6...4
%e A209689 0...1...4...13...5
%e A209689 First three polynomials v(n,x): 1, 2x, 2x + 3x^2.
%t A209689 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209689 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A209689 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209689 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209689 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209689 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209689 TableForm[cu]
%t A209689 Flatten[%]    (* A209689 *)
%t A209689 Table[Expand[v[n, x]], {n, 1, z}]
%t A209689 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209689 TableForm[cv]
%t A209689 Flatten[%]   (* A209690 *)
%Y A209689 Cf. A209690, A208510.
%K A209689 nonn,tabl
%O A209689 1,3
%A A209689 _Clark Kimberling_, Mar 12 2012