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A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

%I A209766 #5 Mar 30 2012 18:58:15
%S A209766 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 A209766 391,239,1,3,13,55,233,739,1113,577,1,3,13,55,233,939,2469,3095,1393,
%U A209766 1,3,13,55,233,987,3589,7903,8457,3363,1,3,13,55,233,987,4085
%N A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.
%C A209766 Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)
%C A209766 Coefficient of x^n in u(n,x): A001333(n)
%C A209766 Row sums:  1,4,11,34,101,304,... A060925.
%C A209766 For a discussion and guide to related arrays, see A208510.
%F A209766 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209766 v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
%F A209766 where u(1,x)=1, v(1,x)=1.
%e A209766 First five rows:
%e A209766 1
%e A209766 1...3
%e A209766 1...3...7
%e A209766 1...3...13...17
%e A209766 1...3...13...43...41
%e A209766 First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^2.
%t A209766 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209766 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209766 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209766 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209766 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209766 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209766 TableForm[cu]
%t A209766 Flatten[%]    (* A209765 *)
%t A209766 Table[Expand[v[n, x]], {n, 1, z}]
%t A209766 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209766 TableForm[cv]
%t A209766 Flatten[%]    (* A209766 *)
%t A209766 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
%t A209766 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
%t A209766 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
%t A209766 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)
%Y A209766 Cf. A209665, A208510.
%K A209766 nonn,tabl
%O A209766 1,3
%A A209766 _Clark Kimberling_, Mar 14 2012