A209583 - OEIS

%I #8 Mar 30 2012 18:58:15

%S 1,1,1,3,3,1,5,10,6,1,9,23,24,10,1,15,51,71,49,15,1,25,104,188,178,90,

%T 21,1,41,205,452,552,390,153,28,1,67,391,1025,1530,1396,776,245,36,1,

%U 109,730,2218,3927,4394,3169,1435,374,45,1,177,1339,4636,9502

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209584; see the Formula section.

%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)+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 five rows:

%e 1

%e 1...1

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

%e 5...10...6....1

%e 9...23...24...10...1

%e First three polynomials u(n,x): 1, 1 + x, 3 + 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_] := (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[%]   (* A209583 *)

%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[%]   (* A209584 *)

%Y Cf. A209584, A208510.

%K nonn,tabl

%O 1,4

%A _Clark Kimberling_, Mar 11 2012