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A156764 Triangle T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)), with T(0, k, m) = 1, b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (-1)^(j+i)*(j+1)*(k+1)^i*StirlingS1(j-1, i) ), b(n, 0) = n!, and m = 3, read by rows.

%I A156764 #7 Jun 21 2021 06:04:57
%S A156764 1,1,1,1,6,1,1,40,40,1,1,300,2000,300,1,1,2520,126000,126000,2520,1,1,
%T A156764 23520,9878400,74088000,9878400,23520,1,1,241920,948326400,
%U A156764 59744563200,59744563200,948326400,241920,1,1,2721600,109734912000,64524128256000,542002677350400,64524128256000,109734912000,2721600,1
%N A156764 Triangle T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)), with T(0, k, m) = 1, b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (-1)^(j+i)*(j+1)*(k+1)^i*StirlingS1(j-1, i) ), b(n, 0) = n!, and m = 3, read by rows.
%H A156764 G. C. Greubel, <a href="/A156764/b156764.txt">Rows n = 0..30 of the triangle, flattened</a>
%F A156764 T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)), with T(0, k, m) = 1, b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (-1)^(j+i)*(j+1)*(k+1)^i*StirlingS1(j-1, i) ), b(n, 0) = n!, and m = 3.
%F A156764 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 = 3. - _G. C. Greubel_, Jun 20 2021
%e A156764 Triangle begins as:
%e A156764   1;
%e A156764   1,      1;
%e A156764   1,      6,         1;
%e A156764   1,     40,        40,           1;
%e A156764   1,    300,      2000,         300,           1;
%e A156764   1,   2520,    126000,      126000,        2520,         1;
%e A156764   1,  23520,   9878400,    74088000,     9878400,     23520,      1;
%e A156764   1, 241920, 948326400, 59744563200, 59744563200, 948326400, 241920, 1;
%t A156764 (* First program *)
%t A156764 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 A156764 T[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
%t A156764 Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 20 2021 *)
%t A156764 (* Second program *)
%t A156764 f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
%t A156764 T[n_, k_, m_]:= If[n==0, 1, f[n,m]/(f[k,m]*f[n-k,m])];
%t A156764 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 20 2021 *)
%o A156764 (Sage)
%o A156764 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 A156764 def T(n,k,m): return 1 if (n==0) else f(n,m)/(f(k,m)*f(n-k,m))
%o A156764 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 20 2021
%Y A156764 Cf. A007318 (m=0), A156584 (m=1), this sequence (m=3).
%K A156764 nonn,tabl
%O A156764 0,5
%A A156764 _Roger L. Bagula_, Feb 15 2009
%E A156764 Edited by _G. C. Greubel_, Jun 20 2021