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%I #7 Sep 08 2022 08:45:41
%S 1,1,1,1,6,1,1,42,42,1,1,360,2520,360,1,1,3720,223200,223200,3720,1,1,
%T 45360,28123200,241056000,28123200,45360,1,1,640080,4839004800,
%U 428597568000,428597568000,4839004800,640080,1,1,10281600,1096841088000,1184588375040000,12240746542080000,1184588375040000,1096841088000,10281600,1
%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 = 1, read by rows.
%H G. C. Greubel, <a href="/A156765/b156765.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 = 1.
%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 = 1. - _G. C. Greubel_, Jun 19 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 42, 42, 1;
%e 1, 360, 2520, 360, 1;
%e 1, 3720, 223200, 223200, 3720, 1;
%e 1, 45360, 28123200, 241056000, 28123200, 45360, 1;
%e 1, 640080, 4839004800, 428597568000, 428597568000, 4839004800, 640080, 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, 1], {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, 1], {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,1): 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,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2021
%Y Cf. A007318 (m=0), this sequence (m=1), A156766 (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
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