A213735 Triangle read by rows: row n is the expansion of x^n in terms of (x+k)!/x! for decreasing k.

1, 1, -1, 1, -3, 1, 1, -6, 7, -1, 1, -10, 25, -15, 1, 1, -15, 65, -90, 31, -1, 1, -21, 140, -350, 301, -63, 1, 1, -28, 266, -1050, 1701, -966, 127, -1, 1, -36, 462, -2646, 6951, -7770, 3025, -255, 1, 1, -45, 750, -5880, 22827, -42525, 34105, -9330, 511, -1

Offset

0,5

Comments

Signed version of A008278.

Links

Table of n, a(n) for n=0..54.

Example

Triangle starts
  1;
  1,  -1;
  1,  -3,  1;
  1,  -6,  7,  -1;
  1, -10, 25, -15,  1;
  1, -15, 65, -90, 31, -1;
  ...
The fourth row corresponds to the expansion x^3 = 1*(x + 3)!/x! - 6*(x + 2)!/x! + 7*(x + 1)!/x! - 1.

Prog

(Maxima) f1(p, n) := if n = 0 then [p] else [coeff(p, x ^ n), f1(p - expand(product(x + i, i, 1, n) * coeff(p, x^ n)), n - 1)]$
f(n) := flatten(f1(x ^ n, n))$

Crossrefs

Sequence in context: A382225 A133713 A008278 * A056858 A137251 A370757

Adjacent sequences: A213732 A213733 A213734 * A213736 A213737 A213738

Keyword

sign,tabl

Author

Douglas Boffey, Jun 18 2012

Status

approved