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A209999 Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.

%I A209999 #5 Mar 30 2012 18:58:16
%S A209999 1,2,2,4,6,3,7,16,13,4,12,36,44,24,5,20,76,122,100,40,6,33,152,306,
%T A209999 332,201,62,7,54,294,712,968,783,370,91,8,88,554,1573,2572,2614,1666,
%U A209999 637,128,9,143,1024,3339,6392,7829,6296,3277,1040,174,10,232,1864
%N A209999 Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.
%C A209999 Column 1: -1+F(n+2), where F=000045 (Fibonacci numbers)
%C A209999 Row sums: A003462
%C A209999 Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...
%C A209999 For a discussion and guide to related arrays, see A208510.
%F A209999 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209999 v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209999 where u(1,x)=1, v(1,x)=1.
%e A209999 First five rows:
%e A209999 1
%e A209999 2....2
%e A209999 4....6....3
%e A209999 7....16...13...4
%e A209999 12...36...44...24...5
%e A209999 First three polynomials u(n,x): 1, 2 + 2x, 4 + 6x + 3x^2.
%t A209999 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209999 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209999 v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209999 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209999 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209999 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209999 TableForm[cu]
%t A209999 Flatten[%]    (* A209999 *)
%t A209999 Table[Expand[v[n, x]], {n, 1, z}]
%t A209999 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209999 TableForm[cv]
%t A209999 Flatten[%]    (* A210287 *)
%Y A209999 Cf. A210287, A208510.
%K A209999 nonn,tabl
%O A209999 1,2
%A A209999 _Clark Kimberling_, Mar 23 2012