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A208608 Triangle of coefficients of polynomials u(n,x) jointly generated with A208609; see the Formula section.

%I A208608 #9 Mar 30 2012 18:58:13
%S A208608 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 A208608 13,42,95,140,136,79,21,1,15,56,148,265,332,279,143,34,1,17,72,217,
%U A208608 455,692,751,558,256,55,1,19,90,304,728,1295,1708,1641,1093,454
%N A208608 Triangle of coefficients of polynomials u(n,x) jointly generated with A208609; see the Formula section.
%C A208608 coefficient of x^(n-1)=Fibonacci(n)=A000045(n)
%F A208608 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208608 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A208608 where u(1,x)=1, v(1,x)=1.
%e A208608 First five rows:
%e A208608 1
%e A208608 1...1
%e A208608 1...3...2
%e A208608 1...5...6....3
%e A208608 1...7...12...12...5
%e A208608 First five polynomials u(n,x):
%e A208608 1
%e A208608 1 + x
%e A208608 1 + 3x + 2x^2
%e A208608 1 + 5x + 6x^2 + 3x^3
%e A208608 1 + 7x + 12x^2 + 12x^3 + 5x^4
%t A208608 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208608 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208608 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A208608 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208608 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208608 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208608 TableForm[cu]
%t A208608 Flatten[%]  (* A208608 *)
%t A208608 Table[Expand[v[n, x]], {n, 1, z}]
%t A208608 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208608 TableForm[cv]
%t A208608 Flatten[%]  (* A208609 *)
%Y A208608 Cf. A208609.
%K A208608 nonn,tabl
%O A208608 1,5
%A A208608 _Clark Kimberling_, Feb 29 2012