This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051714 #53 Jul 02 2025 16:01:58
%S A051714 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 A051714 -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 A051714 -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 A051714 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 A051714 Leading column gives the Bernoulli numbers A164555/A027642. - corrected by _Paul Curtz_, Apr 17 2014
%H A051714 Alois P. Heinz, <a href="/A051714/b051714.txt">Antidiagonals n = 0..140, flattened</a>
%H A051714 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 A051714 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A051714 From _Fabián Pereyra_, Jan 14 2023: (Start)
%F A051714 a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
%F A051714 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 A051714 Table begins:
%e A051714 1 1/2 1/3 1/4 1/5 1/6 1/7 ...
%e A051714 1/2 1/3 1/4 1/5 1/6 1/7 ...
%e A051714 1/6 1/6 3/20 2/15 5/42 ...
%e A051714 0 1/30 1/20 2/35 5/84 ...
%e A051714 -1/30 -1/30 -3/140 -1/105 ...
%e A051714 Antidiagonals of numerator(a(n,k)):
%e A051714 1;
%e A051714 1, 1;
%e A051714 1, 1, 1;
%e A051714 1, 1, 1, 0;
%e A051714 1, 1, 3, 1, -1;
%e A051714 1, 1, 2, 1, -1, 0;
%e A051714 1, 1, 5, 2, -3, -1, 1;
%e A051714 1, 1, 3, 5, -1, -1, 1, 0;
%e A051714 1, 1, 7, 5, 0, -4, 1, 1, -1;
%e A051714 1, 1, 4, 7, 1, -1, -1, 1, -1, 0;
%e A051714 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5;
%p A051714 a:= proc(n,k) option remember;
%p A051714 `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
%p A051714 end:
%p A051714 seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # _Alois P. Heinz_, Apr 17 2013
%t A051714 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 A051714 (Magma)
%o A051714 function a(n,k)
%o A051714 if n eq 0 then return 1/(k+1);
%o A051714 else return (k+1)*(a(n-1,k) - a(n-1,k+1));
%o A051714 end if;
%o A051714 end function;
%o A051714 A051714:= func< n,k | Numerator(a(n,k)) >;
%o A051714 [A051714(k,n-k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Apr 22 2023
%o A051714 (SageMath)
%o A051714 def a(n,k):
%o A051714 if (n==0): return 1/(k+1)
%o A051714 else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
%o A051714 def A051714(n,k): return numerator(a(n, k))
%o A051714 flatten([[A051714(k, n-k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Apr 22 2023
%Y A051714 Rows 2, 3, 4 give: A026741/A045896, A051712/A051713, A051722/A051723.
%Y A051714 Columns 0, 1, 2, 3 give: A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.
%Y A051714 Denominators are in A051715.
%Y A051714 Cf. A027642, A164555.
%K A051714 sign,frac,nice,easy,tabl,look
%O A051714 0,13
%A A051714 _N. J. A. Sloane_
%E A051714 More terms from _James Sellers_, Dec 07 1999