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

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, 8, 15, 19, 27, 35, 44, 37, 47, 9, 17, 22, 32, 43, 57, 71, 60, 76, 10, 19, 25, 37, 51, 70, 92, 115, 97, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 157, 199, 12, 23, 31

Offset: 1

Author

Clark Kimberling, Mar 30 2012

Keywords

• nonn
• tabl

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)

Examples

			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.
		

Crossrefs

Cf. A208510, A210874, A210875.

Programs

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 *)

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.