A051715 - OEIS

%I #48 Feb 06 2023 14:55:12

%S 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,

%T 8,8,28,84,105,28,42,1,9,9,72,84,1,105,140,30,30,10,10,45,120,140,28,

%U 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

%N 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.

%C Leading column gives the Bernoulli numbers A027641/A027642.

%H Alois P. Heinz, <a href="/A051715/b051715.txt">Antidiagonals n = 0..140, flattened</a>

%H M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

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

%e Table begins:

%e     1    1/2   1/3    1/4   1/5  1/6  1/7 ...

%e    1/2   1/3   1/4    1/5   1/6  1/7 ...

%e    1/6   1/6   3/20   2/15  5/42 ...

%e     0    1/30  1/20   2/35  5/84 ...

%e   -1/30 -1/30 -3/140 -1/105 ...

%p a:= proc(n,k) option remember;

%p       `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))

%p     end:

%p seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # _Alois P. Heinz_, Apr 17 2013

%t 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 *)

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

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

%Y Numerators are in A051714.

%K nonn,frac,nice,easy,tabl,look

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Dec 08 1999