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Triangle of coefficients of polynomials u(n,x) jointly generated with A208340; see the Formula section.
2

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

%S 1,2,1,4,5,2,7,14,11,3,11,31,38,22,5,16,60,103,93,43,8,22,106,239,298,

%T 212,81,13,29,175,497,802,782,459,150,21,37,274,952,1909,2393,1917,

%U 958,273,34,46,411,1710,4143,6410,6570,4465,1942,491,55,56,595

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208340; see the Formula section.

%C Every row ends with a Fibonacci number (A000045).

%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*u(n-1,x)+(x+1)*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....5....2

%e 7....14...11...3

%e 11...31...38...22...5

%e First three polynomials v(n,x): 1, 2 + x, 4 + 5x + 2x^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*u[n - 1, x] + (x + 1)*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[%] (* A209153 *)

%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[%] (* A208340 *)

%Y Cf. A208340, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 07 2012