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%I #6 Mar 30 2012 18:58:15
%S 1,1,1,1,3,1,1,3,6,1,1,3,8,10,1,1,3,8,19,15,1,1,3,8,21,40,21,1,1,3,8,
%T 21,53,76,28,1,1,3,8,21,55,125,133,36,1,1,3,8,21,55,142,273,218,45,1,
%U 1,3,8,21,55,144,354,554,339,55,1,1,3,8,21,55,144,375,839,1053
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209564; 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...1
%e 1...3...1
%e 1...3...6...1
%e 1...3...8...10...1
%e First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^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. A209564, A208510.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Mar 10 2012
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