OFFSET
1,3
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)=3x+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 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)
FORMULA
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.
EXAMPLE
First six rows:
1
1...3
3...4....4
4...7....5....7
5...9....10...9....11
6...11...13...17...14...18
First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 4x^2.
MATHEMATICA
u[1, x_] := 1; u[2, x_] := 3 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[%] (* A210881 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 30 2012
STATUS
approved