A144389 - OEIS

A144389

Triangle T(n,k) = n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), T(0,0) = 1, read by rows, 0 <= k <= n.

-1, 2, -1, 1, 4, -1, 4, 3, 6, -1, 3, 16, 6, 8, -1, 6, 15, 40, 10, 10, -1, 5, 36, 45, 80, 15, 12, -1, 8, 35, 126, 105, 140, 21, 14, -1, 7, 64, 140, 336, 210, 224, 28, 16, -1, 10, 63, 288, 420, 756, 378, 336, 36, 18, -1, 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n,k) = [x^k] (n*(x + 1)^(n - 1) - (x - 1)^n).

Sum_{k=0..n} T(n,k) = A001787(n), n >= 1.

EXAMPLE

Triangle begins:

-1;

2, -1;

1, 4, -1;

4, 3, 6, -1;

3, 16, 6, 8, -1;

6, 15, 40, 10, 10, -1;

5, 36, 45, 80, 15, 12, -1;

8, 35, 126, 105, 140, 21, 14, -1;

7, 64, 140, 336, 210, 224, 28, 16, -1;

10, 63, 288, 420, 756, 378, 336, 36, 18, -1;

9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1;

...

MATHEMATICA

p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1);

Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten

PROG

(Maxima) create_list(n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */

CROSSREFS

Cf. A001787, A007318, A130595, A144388, A216973.

Sequence in context: A328649 A281422 A344529 * A136321 A112987 A125138

Adjacent sequences: A144386 A144387 A144388 * A144390 A144391 A144392

KEYWORD

sign,easy,tabl

AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

STATUS

approved