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

%I A209765 #5 Mar 30 2012 18:58:15
%S A209765 1,1,2,1,5,5,1,5,15,12,1,5,21,45,29,1,5,21,77,129,70,1,5,21,89,265,
%T A209765 361,169,1,5,21,89,353,865,991,408,1,5,21,89,377,1325,2717,2681,985,1,
%U A209765 5,21,89,377,1549,4733,8281,7169,2378,1,5,21,89,377,1597,6125
%N A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.
%C A209765 Limiting row: F(2+3k), where F=A000045 (Fibonacci numbers)
%C A209765 Coefficient of x^n in u(n,x): 1,2,5,12,.... A000129(n)
%C A209765 Row sums:  1,3,11,33,101,303,911,2733,..... A081250
%C A209765 Alternating row sums: 1,-1,1,-1,1,-1,,..... A033999
%C A209765 For a discussion and guide to related arrays, see A208510.
%F A209765 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209765 v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
%F A209765 where u(1,x)=1, v(1,x)=1.
%e A209765 First five rows:
%e A209765 1
%e A209765 1...2
%e A209765 1...5...5
%e A209765 1...5...15...12
%e A209765 1...5...21...45...29
%e A209765 First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.
%t A209765 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209765 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209765 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209765 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209765 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209765 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209765 TableForm[cu]
%t A209765 Flatten[%]    (* A209765 *)
%t A209765 Table[Expand[v[n, x]], {n, 1, z}]
%t A209765 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209765 TableForm[cv]
%t A209765 Flatten[%]    (* A209766 *)
%t A209765 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
%t A209765 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
%t A209765 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
%t A209765 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)
%Y A209765 Cf. A209766, A208510.
%K A209765 nonn,tabl
%O A209765 1,3
%A A209765 _Clark Kimberling_, Mar 14 2012