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A209820 Triangle of coefficients of polynomials v(n,x) jointly generated with A209819; see the Formula section.

%I A209820 #6 Mar 30 2012 18:58:16
%S A209820 1,2,2,2,6,5,2,8,18,12,2,8,30,52,29,2,8,34,104,146,70,2,8,34,136,342,
%T A209820 402,169,2,8,34,144,514,1080,1090,408,2,8,34,144,594,1848,3306,2920,
%U A209820 985,2,8,34,144,610,2360,6370,9872,7746,2378,2,8,34,144,610,2552
%N A209820 Triangle of coefficients of polynomials v(n,x) jointly generated with A209819; see the Formula section.
%C A209820 Let T(n,k) be the general term.
%C A209820 T(n,n): A000129
%C A209820 T(n,n-1): 2*A071667
%C A209820 Row sums: A003462
%C A209820 Alternating row sums: 1,0,1,0,1,0,1,0,...
%C A209820 Limiting row: F(3), F(6),F(9),...where F=A000045 (Fibonacci numbers)
%C A209820 For a discussion and guide to related arrays, see A208510.
%F A209820 u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A209820 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209820 where u(1,x)=1, v(1,x)=1.
%e A209820 First five rows:
%e A209820 1
%e A209820 2...2
%e A209820 2...6...5
%e A209820 2...8...18...12
%e A209820 2...8...30...52...29
%e A209820 First three polynomials v(n,x): 1, 2 + 2x , 2 + 6x + 5x^2.
%t A209820 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209820 u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209820 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209820 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209820 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209820 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209820 TableForm[cu]
%t A209820 Flatten[%]    (* A209819 *)
%t A209820 Table[Expand[v[n, x]], {n, 1, z}]
%t A209820 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209820 TableForm[cv]
%t A209820 Flatten[%]    (* A209820 *)
%Y A209820 Cf. A209819, A208510.
%K A209820 nonn,tabl
%O A209820 1,2
%A A209820 _Clark Kimberling_, Mar 23 2012