A209693 - OEIS

%I #6 Mar 30 2012 18:58:15

%S 1,0,2,0,3,3,0,4,9,4,0,5,17,19,5,0,6,27,49,34,6,0,7,39,98,115,55,7,0,

%T 8,53,170,284,236,83,8,0,9,69,269,585,706,440,119,9,0,10,87,399,1070,

%U 1706,1568,763,164,10,0,11,107,564,1799,3577,4395,3193,1250,219

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209694; see the Formula section.

%C Alternating row sums are periodic.

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

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

%F v(n,x)=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 0...2

%e 0...2...4

%e 0...1...8...8

%e 0...1...5...24...16

%e First three polynomials v(n,x): 1, 2x, 2x + 4x^2.

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

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

%t v[n_, x_] := 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[%]   (* A209693 *)

%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[%]   (* A209694 *)

%Y Cf. A209694, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 12 2012