A210880 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

1, 2, 1, 3, 5, 2, 4, 7, 7, 3, 5, 9, 10, 12, 5, 6, 11, 13, 17, 19, 8, 7, 13, 16, 22, 27, 31, 13, 8, 15, 19, 27, 35, 44, 50, 21, 9, 17, 22, 32, 43, 57, 71, 81, 34, 10, 19, 25, 37, 51, 70, 92, 115, 131, 55, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 89, 12, 23, 31, 47

Offset

1,2

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

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

Links

Table of n, a(n) for n=1..70.

Formula

Column k consists of the partial sums of the following sequence: F(k), F(k+2)+F(k-3), F(k+1), F(k+1), F(k+1),..., where F=A000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>1.

Example

First six rows:
  1
  2...1
  3...5....2
  4...7....7....3
  5...9....10...12...5
  6...11...13...17...19...8
First three polynomials u(n,x): 1, 2 + x, 3 + 5x + 2x^2.

Mathematica

u[1, x_] := 1; u[2, x_] := 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[%]   (* A210880 *)

Crossrefs

Cf. A208510, A210881, A210874, A210875.

Sequence in context: A106583 A081450 A248408 * A210867 A019588 A193953

Adjacent sequences: A210877 A210878 A210879 * A210881 A210882 A210883

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 30 2012

Status

approved