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Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.
3

%I #5 Mar 30 2012 18:58:15

%S 1,1,3,1,3,7,1,3,13,17,1,3,13,43,41,1,3,13,55,133,99,1,3,13,55,209,

%T 391,239,1,3,13,55,233,739,1113,577,1,3,13,55,233,939,2469,3095,1393,

%U 1,3,13,55,233,987,3589,7903,8457,3363,1,3,13,55,233,987,4085

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

%C Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)

%C Coefficient of x^n in u(n,x): A001333(n)

%C Row sums: 1,4,11,34,101,304,... A060925.

%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*v(n-1,x)+1,

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

%e First five rows:

%e 1

%e 1...3

%e 1...3...7

%e 1...3...13...17

%e 1...3...13...43...41

%e First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^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*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[%] (* A209765 *)

%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[%] (* A209766 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A081250 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A060925 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

%Y Cf. A209665, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 14 2012