

A210875


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


4



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
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OFFSET

1,4


COMMENTS

Polynomials u(n,k) are defined by u(n,x)=x*u(n1,x)+(x^2)*u(n2,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,n1)*x^(n1).
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), 3F(k1), F(k+2), F(k+1), F(k+1),..., where 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....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


AUTHOR

Clark Kimberling, Mar 30 2012


STATUS

approved



