A210875 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

1, 1, 1, 3, 4, 2, 4, 7, 5, 3, 5, 9, 10, 9, 5, 6, 11, 13, 17, 14, 8, 7, 13, 16, 22, 27, 23, 13, 8, 15, 19, 27, 35, 44, 37, 21, 9, 17, 22, 32, 43, 57, 71, 60, 34, 10, 19, 25, 37, 51, 70, 92, 115, 97, 55, 11, 21, 28, 42, 59, 83, 113, 149, 186, 157, 89, 12, 23, 31, 47, 67

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)

Links

Table of n, a(n) for n=1..71.

Formula

Column k consists of the partial sums of the following sequence: F(k), 3*F(k-1), F(k+2), 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>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

Cf. A208510, A210881, A210874.

Sequence in context: A021296 A323100 A161173 * A238373 A078069 A090131

Adjacent sequences: A210872 A210873 A210874 * A210876 A210877 A210878

Keyword

nonn,tabl,changed

Author

Clark Kimberling, Mar 30 2012

Status

approved