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A210216 Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.

%I A210216 #5 Jul 12 2012 00:40:00
%S A210216 1,1,2,1,3,3,1,3,7,4,1,3,8,14,5,1,3,8,20,25,6,1,3,8,21,46,41,7,1,3,8,
%T A210216 21,54,97,63,8,1,3,8,21,55,133,189,92,9,1,3,8,21,55,143,309,344,129,
%U A210216 10,1,3,8,21,55,144,364,674,591,175,11,1,3,8,21,55,144,376,894
%N A210216 Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.
%C A210216 Limiting row: even-indexed Fibonacci numbers, A001906.
%C A210216 n-th row sum:  -1+2*n
%C A210216 For a discussion and guide to related arrays, see A208510.
%F A210216 u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210216 v(n,x)=xu(n-1,x)+x*v(n-1,x)+1,
%F A210216 where u(1,x)=1, v(1,x)=1.
%e A210216 First five rows:
%e A210216 1
%e A210216 1...2
%e A210216 1...3...3
%e A210216 1...3...7...4
%e A210216 1...3...8...14...5
%e A210216 First three polynomials v(n,x): 1, 1 + 2x , 1 + 3x + 3x^2.
%t A210216 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210216 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210216 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210216 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210216 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210216 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210216 TableForm[cu]
%t A210216 Flatten[%]    (* A210215 *)
%t A210216 Table[Expand[v[n, x]], {n, 1, z}]
%t A210216 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210216 TableForm[cv]
%t A210216 Flatten[%]    (* A210216 *)
%t A210216 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210216 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210216 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
%t A210216 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
%Y A210216 Cf. A210215, A208510.
%K A210216 nonn,tabl
%O A210216 1,3
%A A210216 _Clark Kimberling_, Mar 19 2012