A209767 - OEIS

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

%S 1,1,2,2,6,5,3,12,20,12,4,21,52,63,29,5,33,109,199,187,70,6,48,200,

%T 490,700,536,169,7,66,334,1032,1988,2322,1498,408,8,87,520,1948,4742,

%U 7488,7378,4109,985,9,111,767,3388,10004,19992,26664,22685,11109

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209768; see the Formula section.

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

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

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

%e 2...6....5

%e 3...12...20...12

%e 4...21...52...63...29

%e First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 5x^2.

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

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

%t v[n_, x_] := 2 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[%]    (* A209767 *)

%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[%]    (* A209768 *)

%Y Cf. A209768, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 15 2012