cp's OEIS Frontend

A210874 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

%I A210874 #9 Oct 26 2024 04:57:18
%S A210874 1,2,3,3,5,4,4,7,7,7,5,9,10,12,11,6,11,13,17,19,18,7,13,16,22,27,31,
%T A210874 29,8,15,19,27,35,44,50,47,9,17,22,32,43,57,71,81,76,10,19,25,37,51,
%U A210874 70,92,115,131,123,11,21,28,42,59,83,113,149,186,212,199,12,23,31
%N A210874 Triangular array U(n,k) of coefficients of polynomials defined in Comments.
%C A210874 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+2.  The array (U(n,k)) is defined by rows:
%C A210874 u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
%C A210874 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 A210874 Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)
%F A210874 Column k consists of the partial sums of the following sequence: L(k), F(k+1), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>1.
%e A210874 First six rows:
%e A210874   1
%e A210874   2...3
%e A210874   3...5...4
%e A210874   4...7...7....7
%e A210874   5...9...10...12...11
%e A210874   6...11..13...17...19...18
%e A210874 First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.
%t A210874 u[1, x_] := 1; u[2, x_] := 3 x + 2; z = 14;
%t A210874 u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
%t A210874 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210874 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210874 TableForm[cu]
%t A210874 Flatten[%]     (* A210874 *)
%Y A210874 Cf. A208510, A210881, A210875, A210880.
%K A210874 nonn,tabl
%O A210874 1,2
%A A210874 _Clark Kimberling_, Mar 30 2012