A210876 - OEIS

%I #5 Oct 02 2013 16:26:12

%S 1,2,1,1,5,1,1,4,9,1,1,3,12,14,1,1,3,9,29,20,1,1,3,8,27,60,27,1,1,3,8,

%T 22,74,111,35,1,1,3,8,21,63,181,189,44,1,1,3,8,21,56,178,399,302,54,1,

%U 1,3,8,21,55,154,474,806,459,65,1,1,3,8,21,55,145,430,1169

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.

%C For n>2, each row begins with 1 and ends with 1.  If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).

%C Row sums: A000225 (-1+2^n)

%C Alternating row sums:  A077973

%C Limiting row:  1,3,8,21,55,..., even-indexed Fibonacci numbers

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

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

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

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

%e First six rows:

%e 1

%e 2...1

%e 1...5...1

%e 1...4...9....1

%e 1...3...12...14...1

%e 1...3...9....29...20...1

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

%t u[1, x_] := 1; v[1, x_] := 1; z = 14;

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

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

%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[%]    (* A210876 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A210877 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

%Y Cf. A210877, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 30 2012