A051715 Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.

1, 2, 2, 3, 3, 6, 4, 4, 6, 1, 5, 5, 20, 30, 30, 6, 6, 15, 20, 30, 1, 7, 7, 42, 35, 140, 42, 42, 8, 8, 28, 84, 105, 28, 42, 1, 9, 9, 72, 84, 1, 105, 140, 30, 30, 10, 10, 45, 120, 140, 28, 105, 20, 30, 1, 11, 11, 110, 495, 3960, 924, 231, 165, 220, 66, 66, 12, 12, 66, 55, 495, 264, 308, 132, 165, 44, 66, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• nice
• easy
• tabl
• look

Comments

Leading column gives the Bernoulli numbers A027641/A027642.

Examples

			Table begins:
    1    1/2   1/3    1/4   1/5  1/6  1/7 ...
   1/2   1/3   1/4    1/5   1/6  1/7 ...
   1/6   1/6   3/20   2/15  5/42 ...
    0    1/30  1/20   2/35  5/84 ...
  -1/30 -1/30 -3/140 -1/105 ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Rows 2, 3, 4 give A026741/A045896, A051712/A051713, A051722/A051723.

Columns 0, 1, 2, 3 give A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.

Numerators are in A051714.

Programs

Maple

a:= proc(n,k) option remember;
      `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
    end:
seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013

Mathematica

nmax = 12; a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)(a[n-1, k]-a[n-1, k+1]); Denominator[ Flatten[ Table[ a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]](* Jean-François Alcover, Nov 28 2011 *)

Formula

a(n,k) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)). - Fabián Pereyra, Jan 14 2023

Extensions

More terms from James Sellers, Dec 08 1999