A209564 - OEIS

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

%S 1,1,2,1,2,3,1,2,5,4,1,2,5,11,5,1,2,5,13,21,6,1,2,5,13,32,36,7,1,2,5,

%T 13,34,72,57,8,1,2,5,13,34,87,148,85,9,1,2,5,13,34,89,212,281,121,10,

%U 1,2,5,13,34,89,231,485,499,166,11,1,2,5,13,34,89,233,585,1039

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209559; see the Formula section.

%C A209563:  first k terms of row n are F(2), ..., F(2k), where F = A000045 (Fibonacci numbers) and k=floor ((n+1)/2).

%C A209564:  first k terms of row n are F(1), ..., F(2k-1), where k=floor ((n+2)/2).

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

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

%F v(n,x)=x*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 1...2

%e 1...2...3

%e 1...2...5...4

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

%e First three polynomials v(n,x): 1, 1+2x , 1+2x+3x^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];

%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[%]   (* A209563 *)

%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[%]   (* A209564 *)

%Y Cf. A209563, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 10 2012