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A210874 Triangular array U(n,k) of coefficients of polynomials defined in Comments. 4
1, 2, 3, 3, 5, 4, 4, 7, 7, 7, 5, 9, 10, 12, 11, 6, 11, 13, 17, 19, 18, 7, 13, 16, 22, 27, 31, 29, 8, 15, 19, 27, 35, 44, 50, 47, 9, 17, 22, 32, 43, 57, 71, 81, 76, 10, 19, 25, 37, 51, 70, 92, 115, 131, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 199, 12, 23, 31 (list; table; graph; refs; listen; history; text; internal format)
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)=3x+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 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+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>1.

EXAMPLE

First six rows:

1

2...3

3...5...4

4...7...7.....7

5...9...10....12...11

6...11...13...17...19...18

First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.

MATHEMATICA

u[1, x_] := 1; u[2, x_] := 3 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[%]     (* A210874 *)

CROSSREFS

Cf. A208510, A210881, A210875, A210880.

Sequence in context: A268087 A257004 A126571 * A244796 A080391 A273494

Adjacent sequences:  A210871 A210872 A210873 * A210875 A210876 A210877

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 30 2012

STATUS

approved

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Last modified February 25 11:39 EST 2020. Contains 332233 sequences. (Running on oeis4.)