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Triangle of coefficients of polynomials u(n,x) jointly generated with A209167; see the Formula section.
3

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

%S 1,2,1,4,5,2,7,13,10,3,12,30,34,20,5,20,63,94,80,38,8,33,126,233,258,

%T 177,71,13,54,243,536,728,647,374,130,21,88,457,1171,1881,2043,1527,

%U 765,235,34,143,843,2461,4559,5835,5319,3443,1525,420,55,232

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209167; see the Formula section.

%C Row n begins with F(n+2)-1 and ends with F(n), where F=A000045 (Fibonacci numbers).

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

%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 2....1

%e 5....5....1

%e 11...15...6....1

%e 23...41...27...9...1

%e First three polynomials v(n,x): 1, 2 + x, 5 + 5x + 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. A209167, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 08 2012