A209569 - OEIS

%I #5 Mar 30 2012 18:58:15

%S 1,1,1,2,3,1,3,6,5,1,4,11,14,7,1,5,18,31,26,9,1,6,27,60,71,42,11,1,7,

%T 38,105,162,139,62,13,1,8,51,170,327,372,243,86,15,1,9,66,259,602,861,

%U 754,391,114,17,1,10,83,376,1031,1790,1987,1388,591,146,19,1

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209570; see the Formula section.

%C For n>1, row n begins with n and ends with 2.

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

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

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

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

%e 3...6....5....1

%e 4...11...14...7...1

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

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

%t u[n_, x_] := x*u[n - 1, x] + 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[%]   (* A209569 *)

%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[%]   (* A209570 *)

%Y Cf. A209570, A208510.

%K nonn,tabl

%O 1,4

%A _Clark Kimberling_, Mar 10 2012