A144388 Triangle T(n,k) = binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k), T(0,0) = 1, read by rows, 0 <= k <= n.

1, 0, 1, 3, 0, 1, -2, 9, 0, 1, 5, -8, 18, 0, 1, -4, 25, -20, 30, 0, 1, 7, -24, 75, -40, 45, 0, 1, -6, 49, -84, 175, -70, 63, 0, 1, 9, -48, 196, -224, 350, -112, 84, 0, 1, -8, 81, -216, 588, -504, 630, -168, 108, 0, 1, 11, -80, 405, -720, 1470, -1008, 1050, -240, 135, 0, 1

Offset

0,4

Links

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

Formula

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

Sum_{k=0..n} T(n,k) = A151821(n-1), n >= 1.

Example

Triangle begins:
   1;
   0,   1;
   3,   0,    1;
  -2,   9,    0,    1;
   5,  -8,   18,    0,    1;
  -4,  25,  -20,   30,    0,     1;
   7, -24,   75,  -40,   45,     0,    1;
  -6,  49,  -84,  175,  -70,    63,    0,    1;
   9, -48,  196, -224,  350,  -112,   84,    0,   1;
  -8,  81, -216,  588, -504,   630, -168,  108,   0, 1;
  11, -80,  405, -720, 1470, -1008, 1050, -240, 135, 0, 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(binomial(n, k) + ((-1)^(n + k))*n*binomial(n - 1, k), n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */

Crossrefs

Cf. A001787, A151821, A144389, A216973.

Sequence in context: A263313 A071818 A014513 * A133513 A101000 A035653

Adjacent sequences: A144385 A144386 A144387 * A144389 A144390 A144391

Keyword

sign,easy,tabl

Author

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

Extensions

Edited and offset corrected by Franck Maminirina Ramaharo, Jan 25 2019

Status

approved