A210871 - OEIS

%I #5 Mar 31 2012 20:30:51

%S 1,1,2,1,1,3,1,3,2,5,1,2,7,3,8,1,4,5,15,5,13,1,3,12,10,30,8,21,1,5,9,

%T 31,20,58,13,34,1,4,18,22,73,38,109,21,55,1,6,14,54,51,162,71,201,34,

%U 89,1,5,25,40,145,111,344,130,365,55,144,1,7,20,85,105,361,233

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.

%C Row n, for n>2, starts with 1 and A028242(n) and ends with F(n-1) and F(n+1), where F=A000045 (Fibonacci numbers).

%C Row sums:  A001045

%C Alternating row sums: A077925

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

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

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

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

%e First six rows:

%e 1

%e 1...2

%e 1...1...3

%e 1...3...2....5

%e 1...2...7....3....8

%e 1...4...5....15...5...13

%e First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

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

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

%t v[n_, x_] := (x + 1)*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[%]    (* A210870 *)

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

%t TableForm[cv]

%t Flatten[%]    (* A210871 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

%Y Cf. A210870, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 29 2012