STATUS
proposed
approved
The OEIS is supported by the many generous donors to the OEIS Foundation.
Revision History for A328610
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.
(history; published version)
(history; published version)
STATUS
editing
proposed
NAME
Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.
STATUS
proposed
editing
STATUS
editing
proposed
NAME
Irregular triangular array: the rows show the coefficients of the first of two factors of even -degree polynomials described in Comments.
STATUS
proposed
editing
STATUS
editing
proposed
COMMENTS
It appears that, after the first term, column 1 consists of the Fibonacci numbers, F(k), for k >= 1; see A000045. It appears that after the first row, the row sums are F(2k+1), and the alternating row sums are (-1)^k F(k).
EXAMPLE
*********************
.
-2 , 1 ; (coefficients of -2 + x)
1 , 0 , 1 ; (coefficients of 1 + x^2)
1 , 3 , 0 , 1;
2 , 4 , 6 , 0 , 1;
3 , 10 , 10 , 10 , 0 , 1;
5 , 18 , 30 , 20 , 15 , 0 , 1;
8 , 35 , 63 , 70 , 35 , 21 , 0 , 1;
13 , 64 , 140 , 168 , 140 , 56 , 28 , 0 , 1;
21 , 117 , 288 , 420 , 378 , 252 , 84 , 36 , 0 , 1;
STATUS
proposed
editing
STATUS
editing
proposed
NAME
allocated for Clark KimberlingIrregular triangular array: the rows show the coefficients of the first of two factors of even degree polynomials described in Comments.
DATA
-2, 1, 1, 0, 1, 1, 3, 0, 1, 2, 4, 6, 0, 1, 3, 10, 10, 10, 0, 1, 5, 18, 30, 20, 15, 0, 1, 8, 35, 63, 70, 35, 21, 0, 1, 13, 64, 140, 168, 140, 56, 28, 0, 1, 21, 117, 288, 420, 378, 252, 84, 36, 0, 1, 34, 210, 585, 960, 1050, 756, 420, 120, 45, 0, 1, 55, 374
OFFSET
1,1
COMMENTS
Let p(n) denote the polynomial (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)). It is conjectured in A326925 that if n = 2k, then p(n) = f(k)*g(k), where f(k) and g(k) are polynomials of degree k. Row k of the present array shows the coefficients of f(k).
It appears that, after the first term, column 1 consists of the Fibonacci numbers, F(k), for k >=1; see A000045. It appears that after the first row, the row sums are F(2k+1), and the alternating row sums are (-1)^k F(k).
LINKS
Clark Kimberling, <a href="/A328610/b328610.txt">Table of n, a(n) for n = 1..1325</a>
EXAMPLE
First nine rows:
*********************
-2 1 (coefficients of -2 + x)
1 0 1 (coefficients of 1 + x^2)
1 3 0 1
2 4 6 0 1
3 10 10 10 0 1
5 18 30 20 15 0 1
8 35 63 70 35 21 0 1
13 64 140 168 140 56 28 0 1
21 117 288 420 378 252 84 36 0 1
MATHEMATICA
g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
r[n_] := Rest[f[[n]]];
Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]] (* A328610 *)
Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)
KEYWORD
allocated
sign,tabf
AUTHOR
Clark Kimberling, Oct 24 2019
STATUS
approved
editing