A210597 - OEIS

%I #5 Mar 30 2012 18:58:17

%S 1,2,2,4,5,4,7,13,12,8,12,28,37,28,16,20,58,92,98,64,32,33,114,217,

%T 273,248,144,64,54,218,479,713,760,608,320,128,88,407,1018,1727,2161,

%U 2024,1456,704,256,143,747,2093,3997,5662,6194,5216,3424,1536,512

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210602; see the Formula section.

%C Row n starts with F(n+2)-1, where F=A000045 (Fibonacci

%C numbers), and ends with 2^(n-1).  For a discussion and

%C guide to related arrays, see A208510.

%F u(n,x)=2x*u(n-1,x)+v(n-1,x)+1,

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

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

%e First five rows:

%e 1

%e 2....2

%e 4....5....4

%e 7....13...12...8

%e 12...28...37...28...16

%e First three polynomials u(n,x): 1, 2+ 2x, 4 + 5x + 4x^2.

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

%t u[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;

%t v[n_, x_] := (x + 1)*u[n - 1, 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[%]    (* A210597 *)

%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[%]    (* A210602 *)

%Y Cf. A210602, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 24 2012