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A210881 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

%I A210881 #5 Oct 02 2013 16:26:13
%S A210881 1,1,3,3,4,4,4,7,5,7,5,9,10,9,11,6,11,13,17,14,18,7,13,16,22,27,23,29,
%T A210881 8,15,19,27,35,44,37,47,9,17,22,32,43,57,71,60,76,10,19,25,37,51,70,
%U A210881 92,115,97,123,11,21,28,42,59,83,113,149,186,157,199,12,23,31
%N A210881 Triangular array U(n,k) of coefficients of polynomials defined in Comments.
%C A210881 Polynomials u(n,k) are defined by u(n,x)=x*u(n-1,x)+(x^2)*u(n-2,x)+n*(x+1), where u(1)=1 and u(2,x)=3x+1.  The array (U(n,k)) is defined by rows:
%C A210881 u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
%C A210881 In each column, the first number is a Lucas number and the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
%C A210881 Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)
%F A210881 Column k consists of the partial sums of the following sequence: L(k), F(k-1), F(k+2), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=000045 (Fibonacci numbers).  That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.
%e A210881 First six rows:
%e A210881 1
%e A210881 1...3
%e A210881 3...4....4
%e A210881 4...7....5....7
%e A210881 5...9....10...9....11
%e A210881 6...11...13...17...14...18
%e A210881 First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 4x^2.
%t A210881 u[1, x_] := 1; u[2, x_] := 3 x + 1; z = 14;
%t A210881 u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
%t A210881 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210881 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210881 TableForm[cu]
%t A210881 Flatten[%]   (* A210881 *)
%Y A210881 Cf. A208510, A210874, A210875.
%K A210881 nonn,tabl
%O A210881 1,3
%A A210881 _Clark Kimberling_, Mar 30 2012