A156765 - OEIS

%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