A119502 Triangle read by rows, T(n,k) = (n-k)!, for n>=0 and 0<=k<=n.

1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 24, 6, 2, 1, 1, 120, 24, 6, 2, 1, 1, 720, 120, 24, 6, 2, 1, 1, 5040, 720, 120, 24, 6, 2, 1, 1, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 362880, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 3628800, 362880, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 39916800

Offset: 0

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006

Keywords

• easy
• nonn
• tabl

Comments

The reciprocal of each entry in a lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals one (and all other entries are zero). Note all said entries are unit fractions (all numerators are one).

Denominators of unfinished fractional coefficients for polynomials A152650/A152656 = A009998/A119052. - Paul Curtz, Dec 13 2008

Multiplying the n-th diagonal by b_n with b_0 = 1 and then beheading the triangle provides a Gram matrix whose determinant is related to the reciprocal of e.g.f.s as presented in A133314. - Tom Copeland, Dec 04 2016

Examples

			Triangle starts:
   1;
   1, 1;
   2, 1, 1;
   6, 2, 1, 1;
  24, 6, 2, 1, 1;
		

Crossrefs

Cf. A025581.

Cf. A133314.

Programs

Magma

[[Factorial(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 18 2015

Mathematica

Table[Gamma[Binomial[1 + Floor[(1/2) + Sqrt[2*(1 + n)]], 2] - n], {n, 0, 77}]

Formula

T(n,k) = A025581(n,k)!.

a(n) = Gamma(binomial(1 + floor((1/2) + sqrt(2*(1 + n))), 2) - n).

Extensions

Name edited by Peter Luschny, Jun 17 2015