%I #6 Jul 12 2012 00:40:00
%S 1,3,1,4,3,2,5,6,8,3,6,10,18,14,5,7,15,33,38,27,8,8,21,54,81,83,49,13,
%T 9,28,82,150,197,170,89,21,10,36,118,253,401,448,342,159,34,11,45,163,
%U 399,736,999,987,671,282,55,12,55,218,598,1253,1988,2387,2106
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.
%C For n>1, row n starts with n+1, followed by the n-th
%C triangular number, and ends in F(n+1), where F=A000045
%C (Fibonacci numbers).
%C Column 3: A166830.
%C Row sums: A048654.
%C Alternating row sums: 1,2,3,4,5,6,7,8,9,...
%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),
%F v(n,x)=(x+1)*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 3...1
%e 4...3....2
%e 5...6....8....3
%e 6...10...18...14...5
%e First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 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];
%t v[n_, x_] := (x + 1)*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[%] (* A209703 *)
%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[%] (* A209704 *)
%Y Cf. A209703, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 12 2012