A210869 - OEIS

%I #13 Dec 26 2021 04:40:16

%S 1,0,2,1,0,3,0,3,0,5,1,0,7,0,8,0,4,0,15,0,13,1,0,12,0,30,0,21,0,5,0,

%T 31,0,58,0,34,1,0,18,0,73,0,109,0,55,0,6,0,54,0,162,0,201,0,89,1,0,25,

%U 0,145,0,344,0,365,0,144,0,7,0,85,0,361,0,707,0,655,0,233,1,0

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210868; see the Formula section.

%C Row n starts with 1 or 0 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Row sums:  1,2,4,8,16,32,... (A000079)

%C Alternating row sums: 1, -2, 4, -8, 16,... (A122803)

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

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

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

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

%F T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Apr 02 2012

%e First six rows:

%e 1

%e 0...2

%e 1...0...3

%e 0...3...0...5

%e 1...0...7...0....8

%e 0...4...0...15...0...13

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

%t u[1,x_]:=1;v[1,x_]:=1;z=14;

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

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

%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[%]   (* A210868 *)

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

%t TableForm[cv]

%t Flatten[%]   (* A210869 *)

%t Table[u[n,x]/.x->1,{n,1,z}]   (* A000079 *)

%t Table[v[n,x]/.x->1,{n,1,z}]   (* A000079 *)

%t Table[u[n,x]/.x->-1,{n,1,z}]  (* A151575 *)

%t Table[v[n,x]/.x->-1,{n,1,z}]  (* A122803 *)

%Y Cf. A210868, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 29 2012