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

%S 1,1,2,2,5,4,3,9,13,8,4,15,31,35,16,5,23,61,97,85,32,6,33,107,219,279,

%T 203,64,7,45,173,433,717,761,469,128,8,59,263,779,1583,2195,1991,1067,

%U 256,9,75,381,1305,3141,5361,6381,5049,2389,512,10,93,531

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.

%C Row n begins with n and ends with 2^(n-1).

%C Row sums: 1,3,11,33,101,303,911,... A081250

%C Alternating row sums: 1,-1,1,-1,1,.. A033999

%C For a discussion and guide to related arrays, see A208510.

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

%F v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 1...2

%e 2...5....4

%e 3...9....13...8

%e 4...15...31...35...16

%e First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.

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

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

%t v[n_, x_] := 2 x*u[n - 1, 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[%] (* A209763 *)

%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[%] (* A209764 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A081250 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A060925 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442 *)

%Y Cf. A209764, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 14 2012