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

%I A210215 #5 Jul 12 2012 00:40:00
%S A210215 1,2,1,2,4,1,2,5,7,1,2,5,12,11,1,2,5,13,26,16,1,2,5,13,33,51,22,1,2,5,
%T A210215 13,34,79,92,29,1,2,5,13,34,88,176,155,37,1,2,5,13,34,89,221,365,247,
%U A210215 46,1,2,5,13,34,89,232,530,709,376,56,1,2,5,13,34,89,233,596
%N A210215 Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.
%C A210215 Limiting row:  odd-indexed Fibonacci numbers, (A122367, A001519)
%C A210215 n-th row sum: -1+2^n
%C A210215 For a discussion and guide to related arrays, see A208510.
%F A210215 u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210215 v(n,x)=xu(n-1,x)+x*v(n-1,x)+1,
%F A210215 where u(1,x)=1, v(1,x)=1.
%e A210215 First five rows:
%e A210215 1
%e A210215 2...1
%e A210215 2...4...1
%e A210215 2...5...7....1
%e A210215 2...5...12...11...1
%e A210215 First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^2.
%t A210215 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210215 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210215 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210215 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210215 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210215 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210215 TableForm[cu]
%t A210215 Flatten[%]    (* A210215 *)
%t A210215 Table[Expand[v[n, x]], {n, 1, z}]
%t A210215 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210215 TableForm[cv]
%t A210215 Flatten[%]    (* A210216 *)
%t A210215 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210215 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210215 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
%t A210215 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
%Y A210215 Cf. A210216, A208510.
%K A210215 nonn,tabl
%O A210215 1,2
%A A210215 _Clark Kimberling_, Mar 19 2012