A210215 - OEIS

%I #5 Jul 12 2012 00:40:00

%S 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 13,34,79,92,29,1,2,5,13,34,88,176,155,37,1,2,5,13,34,89,221,365,247,

%U 46,1,2,5,13,34,89,232,530,709,376,56,1,2,5,13,34,89,233,596

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

%C Limiting row:  odd-indexed Fibonacci numbers, (A122367, A001519)

%C n-th row sum: -1+2^n

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=x*u(n-1,x)+v(n-1,x)+1,

%F v(n,x)=xu(n-1,x)+x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 2...1

%e 2...4...1

%e 2...5...7....1

%e 2...5...12...11...1

%e First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;

%t v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A210215 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A210216 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

%Y Cf. A210216, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 19 2012