%I #5 Mar 30 2012 18:58:16
%S 1,2,1,3,2,2,4,3,5,3,5,4,9,8,5,6,5,14,15,15,8,7,6,20,24,31,26,13,8,7,
%T 27,35,54,57,46,21,9,8,35,48,85,104,108,80,34,10,9,44,63,125,170,209,
%U 199,139,55,11,10,54,80,175,258,360,404,366,240,89,12,11,65,99
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.
%C Let T(n,k) denote the term in row n, column k.
%C T(n,n): A000045 (Fibonacci numbers)
%C T(n,n-1): A006367
%C T(n,n-2): A105423
%C T(n,1): 1,2,3,4,5,6,7,8,9,...
%C T(n,2): 1,2,3,4,5,6,7,8,9,...
%C T(n,3): A000096
%C T(n,4): A005563
%C T(n,5): A055831
%C T(n,6): A111694
%C Row sums: A000225
%C Alternating row sums: A052551
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 2...1
%e 3...2...2
%e 4...3...5...3
%e 5...4...9...8...5
%e First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t v[n_, x_] := x*u[n - 1, 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[%] (* A210552 *)
%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[%] (* A210553 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
%Y Cf. A210552, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 22 2012