%I #7 Sep 08 2022 08:45:41
%S 1,1,1,1,8,1,1,78,78,1,1,960,9360,960,1,1,14520,1742400,1742400,14520,
%T 1,1,262080,475675200,5854464000,475675200,262080,1,1,5508720,
%U 180465667200,33594378048000,33594378048000,180465667200,5508720,1,1,132249600,91065752064000,305980926935040000,4627961519892480000,305980926935040000,91065752064000,132249600,1
%N Triangle T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 2, read by rows.
%H G. C. Greubel, <a href="/A156766/b156766.txt">Rows n = 0..30 of the triangle, flattened</a>
%F T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 2.
%F T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = (-1/k)^n * BarnesG(n+2) * q-Pochhammer(k+1, k+1, n) and m = 2. - _G. C. Greubel_, Jun 19 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 78, 78, 1;
%e 1, 960, 9360, 960, 1;
%e 1, 14520, 1742400, 1742400, 14520, 1;
%e 1, 262080, 475675200, 5854464000, 475675200, 262080, 1;
%t (* First program *)
%t b[n_, k_]:= If[k==0, n!, Product[j!*((k+1)^j -1)/k, {j, n}]];
%t T[n_, k_, m_]:= If[n==0, 1, b[n,m]/(b[k,m]*b[n-k,m])];
%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 19 2021 *)
%t (* Second program *)
%t f[n_, k_]:= If[k==0, n!, (-1)^n*BarnesG[n+2] QPochhammer[k+1, k+1, n]/k^n];
%t T[n_, k_, m_]:= If[n==0, 1, f[n,m]/(f[k,m]*f[n-k,m])];
%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 19 2021 *)
%o (Magma)
%o f:= func< n,k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[Factorial(j)*((k+1)^j-1): j in [1..n]])/k^n >;
%o T:= func< n,k,m | n eq 0 select 1 else f(n,m)/(f(k,m)*f(n-k,m)) >;
%o [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2021
%o (Sage)
%o from sage.combinat.q_analogues import q_pochhammer
%o @CachedFunction
%o def f(n,k): return factorial(n) if (k==0) else (-1/k)^n*product( factorial(j) for j in (1..n) )*q_pochhammer(n,k+1,k+1)
%o def T(n,k,m): return 1 if (n==0) else f(n,m)/(f(k,m)*f(n-k,m))
%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2021
%Y Cf. A007318 (m=0), A156765 (m=1), this sequence (m=2), A156767 (m=3).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 15 2009
%E Edited by _G. C. Greubel_, Jun 19 2021