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Triangle of coefficients of polynomials u(n,x) jointly generated with A208609; see the Formula section.
3

%I #9 Mar 30 2012 18:58:13

%S 1,1,1,1,3,2,1,5,6,3,1,7,12,12,5,1,9,20,29,23,8,1,11,30,56,64,43,13,1,

%T 13,42,95,140,136,79,21,1,15,56,148,265,332,279,143,34,1,17,72,217,

%U 455,692,751,558,256,55,1,19,90,304,728,1295,1708,1641,1093,454

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208609; see the Formula section.

%C coefficient of x^(n-1)=Fibonacci(n)=A000045(n)

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

%e 1...3...2

%e 1...5...6....3

%e 1...7...12...12...5

%e First five polynomials u(n,x):

%e 1

%e 1 + x

%e 1 + 3x + 2x^2

%e 1 + 5x + 6x^2 + 3x^3

%e 1 + 7x + 12x^2 + 12x^3 + 5x^4

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

%t u[n_, x_] := u[n - 1, x] + x*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[%] (* A208608 *)

%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[%] (* A208609 *)

%Y Cf. A208609.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 29 2012