A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769

Offset: 0

Author

N. J. A. Sloane, Apr 18 2000

Keywords

• sign
• tabl
• easy
• nice

Examples

			Triangle begins:
  1;
  1,  -1;
  1,  -5,    6;
  1, -12,   47,    -60;
  1, -22,  179,   -638,    840;
  1, -35,  485,  -3325,  11274,   -15120;
  1, -51, 1075, -11985,  74524,  -245004,  332640;
  1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened).
• Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

Crossrefs

Cf. A000407, A054649, A054651, A054654, A008276.

Programs

Maple

a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010

Mathematica

row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Apr 06 2012, after Maple *)

PARI

T(n,k)=polcoef(n!*binomial(x-n,n), n-k);

Formula

n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).

From Robert Israel, Jul 12 2016: (Start)

G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).

E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)