OFFSET

1,4

COMMENTS

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)=x+1. 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 Fibonacci number and, with one exception, the difference between each two consecutive terms is a Fibonacci number (see the Formula section).

Alternating row sums: 1,0,1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

FORMULA

Column k consists of the partial sums of the following sequence: F(k), 3F(k-1), F(k+2), F(k+1), F(k+1),..., where F=000045 (Fibonacci numbers. That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.

EXAMPLE

First six rows:

1

1...1

3...4....2

4...7....5....3

5...9....10...9....5

6...11...13...17...14...8

First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 2x^2.

MATHEMATICA

u[1, x_] := 1; u[2, x_] := x + 1; 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[%] (* A210875 *)

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 30 2012

STATUS

approved