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Triangle of coefficients of polynomials u(n,x) jointly generated with A209169; see the Formula section.
3

%I #5 Mar 30 2012 18:58:15

%S 1,2,1,4,6,3,7,16,17,7,12,37,56,47,17,20,78,154,182,128,41,33,156,378,

%T 574,565,344,99,54,301,864,1590,1995,1697,915,239,88,566,1877,4048,

%U 6118,6605,4973,2413,577,143,1044,3927,9693,17073,22128,21093

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209169; see the Formula section.

%C Row n begins with F(n+2)-1, 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)+2x*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 4....6....3

%e 7....16...17...7

%e 12...37...56...47...17

%e First three polynomials v(n,x): 1, 2 + x, 4 + 6x + 3x^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] + 2 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[%] (* A209168 *)

%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[%] (* A209169 *)

%Y Cf. A209169, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 08 2012