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Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.
4

%I #5 Mar 30 2012 18:58:16

%S 1,2,2,4,6,3,7,16,13,4,12,36,44,24,5,20,76,122,100,40,6,33,152,306,

%T 332,201,62,7,54,294,712,968,783,370,91,8,88,554,1573,2572,2614,1666,

%U 637,128,9,143,1024,3339,6392,7829,6296,3277,1040,174,10,232,1864

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.

%C Column 1: -1+F(n+2), where F=000045 (Fibonacci numbers)

%C Row sums: A003462

%C Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...

%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)+1,

%F v(n,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....2

%e 4....6....3

%e 7....16...13...4

%e 12...36...44...24...5

%e First three polynomials u(n,x): 1, 2 + 2x, 4 + 6x + 3x^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] + 1;

%t v[n_, 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[%] (* A209999 *)

%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[%] (* A210287 *)

%Y Cf. A210287, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 23 2012