login
Triangle of coefficients of polynomials v(n,x) jointly generated with A210795; see the Formula section.
3

%I #5 Mar 30 2012 18:58:17

%S 1,1,2,3,3,3,3,7,6,5,5,10,16,12,8,5,16,26,34,23,13,7,21,47,64,70,43,

%T 21,7,29,68,123,147,140,79,34,9,36,104,200,304,324,274,143,55,9,46,

%U 140,324,538,714,690,527,256,89,11,55,195,480,932,1366,1616,1431

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210795; see the Formula section.

%C Row n starts with A109613(n) and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Column 2: A114113

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

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

%F v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 1...2

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

%e 3...7....6....5

%e 5...10...16...12...8

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

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

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

%t d[x_] := h + x; e[x_] := p + x;

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

%t j = 0; c = 1; h = 2; p = -1; f = 0;

%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[%] (* A210795 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210796 *)

%Y Cf. A210795, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 26 2012