A210880 -id:A210880 - cp's OEIS frontend

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

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, 29, 8, 15, 19, 27, 35, 44, 50, 47, 9, 17, 22, 32, 43, 57, 71, 81, 76, 10, 19, 25, 37, 51, 70, 92, 115, 131, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 199, 12, 23, 31

Offset: 1

Clark Kimberling, Mar 30 2012

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:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
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).
Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

			First six rows:
  1
  2...3
  3...5...4
  4...7...7....7
  5...9...10...12...11
  6...11..13...17...19...18
First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.

Cf. A208510, A210881, A210875, A210880.

u[1, x_] := 1; u[2, x_] := 3 x + 2; z = 14;
u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* 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.

cp's OEIS Frontend

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

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