login
Triangle of coefficients of polynomials u(n,x) jointly generated with A210560; see the Formula section.
4

%I #5 Mar 30 2012 18:58:16

%S 1,1,2,1,4,3,1,6,8,5,1,8,15,17,8,1,10,24,38,33,13,1,12,35,70,86,63,21,

%T 1,14,48,115,180,187,117,34,1,16,63,175,330,437,390,214,55,1,18,80,

%U 252,553,882,1007,791,386,89,1,20,99,348,868,1610,2219,2235,1567

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210560; 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*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 1...2

%e 1...4...3

%e 1...6...8....5

%e 1...8...15...17...8

%e First three polynomials u(n,x): 1, 1 + 2x, 1 + 4x + 3x^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] + 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[%] (* A210559 *)

%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[%] (* A210560 *)

%Y Cf. A210560, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 22 2012