A054649 Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.

1, 1, 0, 1, -3, 4, 1, -9, 32, -36, 1, -18, 131, -426, 528, 1, -30, 375, -2370, 7544, -9600, 1, -45, 865, -8955, 52414, -163800, 213120, 1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240, 1, -84, 3122, -66696, 893249, -7664916, 41096908, -125747664, 167973120

Offset

0,5

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

T(n, k) = n! * [x^(n - k)] hypergeom([-n, -x + n], [-n], -1). - Peter Luschny, Nov 27 2021

Example

Triangle begins:
  1;
  1,   0;
  1,  -3,    4;
  1,  -9,   32,    -36;
  1, -18,  131,   -426,    528;
  1, -30,  375,  -2370,   7544,    -9600;
  1, -45,  865,  -8955,  52414,  -163800,  213120;
  1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240;
  ...
From Peter Luschny, Nov 27 2021: (Start)
The row reversed triangle can be seen as the coefficients of a sequence of monic polynomials with monomials sorted in ascending order which start:
[0]     1;
[1]              x;
[2]     4 -    3*x +      x^2;
[3]   -36 +   32*x -    9*x^2 +     x^3;
[4]   528 -  426*x +  131*x^2 -  18*x^3 +    x^4;
[5] -9600 + 7544*x - 2370*x^2 + 375*x^3 - 30*x^4 + x^5; (End)

Maple

# Some older Maple versions are known to have a bug in the hypergeom function.
with(ListTools): with(PolynomialTools):
CoeffList := p -> op(Reverse(CoefficientList(simplify(p), x))):
p := k -> k!*hypergeom([-k, -x + k], [-k], -1):
seq(CoeffList(p(k)), k = 0..8); # Peter Luschny, Nov 27 2021

Mathematica

c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x-n, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 8}] // Flatten  (* Jean-François Alcover, Oct 04 2012 *)

Prog

(PARI) row(n) = Vec(n!*sum(k=0, n, binomial(x-n, k))); \\ Seiichi Manyama, Sep 24 2021

Crossrefs

Cf. A008275, A008276, A048994, A054651, A054655.

Sequence in context: A164948 A102051 A078068 * A138263 A147721 A172094

Adjacent sequences: A054646 A054647 A054648 * A054650 A054651 A054652

Keyword

sign,tabl,nice

Author

N. J. A. Sloane, Apr 16 2000

Status

approved