%I #52 Apr 23 2023 07:29:23

%S 1,1,1,1,1,1,1,1,1,0,1,1,3,1,-1,1,1,2,1,-1,0,1,1,5,2,-3,-1,1,1,1,3,5,

%T -1,-1,1,0,1,1,7,5,0,-4,1,1,-1,1,1,4,7,1,-1,-1,1,-1,0,1,1,9,28,49,-29,

%U -5,8,1,-5,5,1,1,5,3,8,-7,-9,5,7,-5,5,0,1,1,11,15,27,-28,-343,295,200,-44,-1017,691,-691

%N Numerators 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 A164555/A027642. - corrected by _Paul Curtz_, Apr 17 2014

%H Alois P. Heinz, <a href="/A051714/b051714.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 From _Fabián Pereyra_, Jan 14 2023: (Start)

%F a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).

%F E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)

%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 ...

%e Antidiagonals of numerator(a(n,k)):

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 1, 0;

%e 1, 1, 3, 1, -1;

%e 1, 1, 2, 1, -1, 0;

%e 1, 1, 5, 2, -3, -1, 1;

%e 1, 1, 3, 5, -1, -1, 1, 0;

%e 1, 1, 7, 5, 0, -4, 1, 1, -1;

%e 1, 1, 4, 7, 1, -1, -1, 1, -1, 0;

%e 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5;

%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(numer(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]); Numerator[Flatten[Table[a[n-k, k], {n,0,nmax}, {k, n, 0, -1}]]] (* _Jean-François Alcover_, Nov 28 2011 *)

%o (Magma)

%o function a(n,k)

%o if n eq 0 then return 1/(k+1);

%o else return (k+1)*(a(n-1,k) - a(n-1,k+1));

%o end if;

%o end function;

%o A051714:= func< n,k | Numerator(a(n,k)) >;

%o [A051714(k,n-k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Apr 22 2023

%o (SageMath)

%o def a(n,k):

%o if (n==0): return 1/(k+1)

%o else: return (k+1)*(a(n-1, k) - a(n-1, k+1))

%o def A051714(n,k): return numerator(a(n, k))

%o flatten([[A051714(k, n-k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Apr 22 2023

%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 Denominators are in A051715.

%Y Cf. A027642, A164555.

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

%O 0,13

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Dec 07 1999