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Triangle of coefficients of polynomials v(n,x) jointly generated with A209689; see the Formula section.
3

%I #8 Apr 05 2012 18:37:11

%S 1,2,1,1,4,1,1,3,7,1,1,2,9,11,1,1,2,6,22,16,1,1,2,5,19,46,22,1,1,2,5,

%T 14,54,86,29,1,1,2,5,13,42,135,148,37,1,1,2,5,13,35,124,302,239,46,1,

%U 1,2,5,13,34,99,341,617,367,56,1,1,2,5,13,34,90,287,860,1171

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209689; see the Formula section.

%C Combinatorial limit of rows: odd-indexed Fibonacci numbers. For a discussion and guide to related arrays, see A208510.

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

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

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

%e First five rows:

%e 1

%e 2...1

%e 1...4...1

%e 1...3...7...1

%e 1...2...9...11...1

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

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

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

%t v[n_, 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[%] (* A209689 *)

%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[%] (* A209690 *)

%Y Cf. A209689, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 12 2012