A210743 - OEIS

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

%S 1,2,3,4,8,7,7,20,25,17,12,43,76,75,41,20,88,194,264,216,99,33,172,

%T 458,770,861,606,239,54,327,1016,2038,2811,2691,1667,577,88,608,2161,

%U 5012,8206,9689,8149,4517,1393,143,1112,4447,11699,22057,30830

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210744; see the Formula section.

%C Row n starts with -1+F(n), where F=A000045 (Fibonacci numbers), and ends with A001333(n).  For a discussion and guide to related arrays, see A208510.

%F u(n,x)=2x*u(n-1,x)+(x+1)*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....3

%e 4....8....7

%e 7....20...25...17

%e 12...43...76...75...41

%e First three polynomials u(n,x): 1, 2+ 3x, 4 + 8x + 7x^2.

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

%t u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]      (* A210743 *)

%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[%]      (* A210744 *)

%Y Cf. A210744, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 24 2012