A209167 - OEIS

%I #6 Mar 30 2012 18:58:15

%S 1,2,2,3,5,3,5,12,12,5,8,25,35,25,8,13,50,89,89,50,13,21,96,207,263,

%T 207,96,21,34,180,455,698,698,455,180,34,55,331,959,1719,2073,1719,

%U 959,331,55,89,600,1959,4011,5643,5643,4011,1959,600,89,144,1075

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209166; see the Formula section.

%C Row n begins and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,1,0,...

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

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

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

%e 6....4....1

%e 12...14...7....1

%e 24...40...28...8...1

%t First three polynomials v(n,x): 1, 3 + x, 6 + 4x + x^2.

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

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

%t v[n_, x_] := (x + 1)*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[%]    (* A209166 *)

%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[%]    (* A209167 *)

%Y Cf. A209166, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 08 2012