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 A156584 #19 Jun 21 2021 06:03:26
%S A156584 1,1,1,1,3,1,1,12,12,1,1,60,240,60,1,1,360,7200,7200,360,1,1,2520,
%T A156584 302400,1512000,302400,2520,1,1,20160,16934400,508032000,508032000,
%U A156584 16934400,20160,1,1,181440,1219276800,256048128000,1536288768000,256048128000,1219276800,181440,1
%N A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.
%H A156584 G. C. Greubel, <a href="/A156584/b156584.txt">Rows n = 0..30 of the triangle, flattened</a>
%F A156584 From _G. C. Greubel_, Jun 21 2021: (Start)
%F A156584 T(n, k) = BarnesG(n+3)/(BarnesG(k+3)*BarnesG(n-k+3)).
%F A156584 T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), with T(0, k, m) = 1, f(n, k) = (-1)^n*(n + 1)!*BarnesG(n+k+1)/(Gamma(k+1)^n*BarnesG(k+1)), f(n, 0) = n!, and m = 1. (End)
%e A156584 Triangle begins as:
%e A156584 1;
%e A156584 1, 1;
%e A156584 1, 3, 1;
%e A156584 1, 12, 12, 1;
%e A156584 1, 60, 240, 60, 1;
%e A156584 1, 360, 7200, 7200, 360, 1;
%e A156584 1, 2520, 302400, 1512000, 302400, 2520, 1;
%e A156584 1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1;
%p A156584 SF := n -> mul(j!, j=0..n): T := (n,k) -> SF(n-1)/(SF(n-k)*SF(k)):
%p A156584 seq(print(seq(T(n,k),k=1..n-1)),n=0..9); # _Peter Luschny_, Jan 24 2015
%t A156584 (* First program *)
%t A156584 b[n_, k_]:= If[k==0, n!, Product[Sum[(-1)^(i+j)*(j+1)*StirlingS1[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, 1, n}]];
%t A156584 T[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
%t A156584 Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 20 2021 *)
%t A156584 (* Second program *)
%t A156584 f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
%t A156584 T[n_, k_, m_]:= If[n==0, 1, f[n,m]/(f[k,m]*f[n-k,m])];
%t A156584 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 20 2021 *)
%o A156584 (Sage)
%o A156584 def f(n,k): return factorial(n) if (k==0) else (-1)^n*factorial(n+1)*product( rising_factorial(k+1, j) for j in (0..n-1) )
%o A156584 def T(n,k,m): return 1 if (n==0) else f(n,m)/(f(k,m)*f(n-k,m))
%o A156584 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 21 2021
%Y A156584 Cf. A007318 (m=0), this sequence (m=1), A156764 (m=3).
%Y A156584 Cf. A009963.
%K A156584 nonn,tabl,easy
%O A156584 0,5
%A A156584 _Roger L. Bagula_, Feb 10 2009
%E A156584 New name and editing, _Peter Luschny_, Jan 24 2015