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

1,3

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)

Links

Table of n, a(n) for n=1..69.

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

Cf. A208510, A210874, A210875.

Sequence in context: A243348 A136546 A278765 * A058729 A021303 A303821

Adjacent sequences: A210878 A210879 A210880 * A210882 A210883 A210884

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 30 2012

Status

approved