OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..30 of the triangle, flattened
FORMULA
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.
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
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 42, 42, 1;
1, 360, 2520, 360, 1;
1, 3720, 223200, 223200, 3720, 1;
1, 45360, 28123200, 241056000, 28123200, 45360, 1;
1, 640080, 4839004800, 428597568000, 428597568000, 4839004800, 640080, 1;
MATHEMATICA
(* First program *)
b[n_, k_]:= If[k==0, n!, Product[j!*((k+1)^j -1)/k, {j, n}]];
T[n_, k_, m_]:= If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 19 2021 *)
(* Second program *)
f[n_, k_]:= If[k==0, n!, (-1)^n*BarnesG[n+2] QPochhammer[k+1, k+1, n]/k^n];
T[n_, k_, m_]:= If[n==0, 1, f[n, m]/(f[k, m]*f[n-k, m])];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)
PROG
(Magma)
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 >;
T:= func< n, k, m | n eq 0 select 1 else f(n, m)/(f(k, m)*f(n-k, m)) >;
[T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021
(Sage)
from sage.combinat.q_analogues import q_pochhammer
@CachedFunction
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)
def T(n, k, m): return 1 if (n==0) else f(n, m)/(f(k, m)*f(n-k, m))
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 15 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 19 2021
STATUS
approved