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A209563 Triangle of coefficients of polynomials u(n,x) jointly generated with A209564; see the Formula section.

%I A209563 #6 Mar 30 2012 18:58:15
%S A209563 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 A209563 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 A209563 1,3,8,21,55,144,354,554,339,55,1,1,3,8,21,55,144,375,839,1053
%N A209563 Triangle of coefficients of polynomials u(n,x) jointly generated with A209564; see the Formula section.
%C A209563 A209563:  first k terms of row n are F(2),...,F(2k), where F = A000045 (Fibonacci numbers) and k=floor ((n+1)/2).
%C A209563 A209564:  first k terms of row n are F(1), ..., F(2k-1), where k=floor ((n+2)/2).
%C A209563 For a discussion and guide to related arrays, see A208510.
%F A209563 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209563 v(n,x)=x*u(n-1,x)+x*v(n-1,x) +1,
%F A209563 where u(1,x)=1, v(1,x)=1.
%e A209563 First five rows:
%e A209563 1
%e A209563 1...1
%e A209563 1...3...1
%e A209563 1...3...6...1
%e A209563 1...3...8...10...1
%e A209563 First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
%t A209563 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209563 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209563 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209563 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209563 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209563 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209563 TableForm[cu]
%t A209563 Flatten[%]   (* A209563 *)
%t A209563 Table[Expand[v[n, x]], {n, 1, z}]
%t A209563 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209563 TableForm[cv]
%t A209563 Flatten[%]   (* A209564 *)
%Y A209563 Cf. A209564, A208510.
%K A209563 nonn,tabl
%O A209563 1,5
%A A209563 _Clark Kimberling_, Mar 10 2012