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%I #6 Mar 30 2012 18:58:15
%S 1,1,2,1,2,3,1,2,5,4,1,2,5,11,5,1,2,5,13,21,6,1,2,5,13,32,36,7,1,2,5,
%T 13,34,72,57,8,1,2,5,13,34,87,148,85,9,1,2,5,13,34,89,212,281,121,10,
%U 1,2,5,13,34,89,231,485,499,166,11,1,2,5,13,34,89,233,585,1039
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209559; see the Formula section.
%C A209563: first k terms of row n are F(2), ..., F(2k), where F = A000045 (Fibonacci numbers) and k=floor ((n+1)/2).
%C A209564: first k terms of row n are F(1), ..., F(2k-1), where k=floor ((n+2)/2).
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=x*u(n-1,x)+v(n-1,x),
%F v(n,x)=x*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...2
%e 1...2...3
%e 1...2...5...4
%e 1...2...5...11...1
%e First three polynomials v(n,x): 1, 1+2x , 1+2x+3x^2 .
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := 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[%] (* A209563 *)
%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[%] (* A209564 *)
%Y Cf. A209563, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 10 2012
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