A210198 - OEIS

%I #6 Oct 17 2012 10:05:03

%S 1,3,1,7,4,15,12,1,31,32,6,63,80,24,1,127,192,80,8,255,448,240,40,1,

%T 511,1024,672,160,10,1023,2304,1792,560,60,1,2047,5120,4608,1792,280,

%U 12,4095,11264,11520,5376,1120,84,1,8191,24576,28160,15360,4032

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210197; see the Formula section.

%C Row sums:  A005409

%C Column 1:  -1+2^n

%C Alternating row sums: 1, 2,3,4,5,6,..., A000027

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

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

%F v(n,x)=(x+1)*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 3....1

%e 15...12...1

%e 31...32...6

%e 63...80...24...1

%e First three polynomials v(n,x): 1, 3 + x , 15 + 12x + x^2.

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

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

%t v[n_, x_] := (x + 1)*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[%]   (* A210197 *)

%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[%]  (* A210198 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A048739 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A005409 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)

%Y Cf. A210197, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 18 2012