OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..30 of the triangle, flattened
FORMULA
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.
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
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 40, 40, 1;
1, 300, 2000, 300, 1;
1, 2520, 126000, 126000, 2520, 1;
1, 23520, 9878400, 74088000, 9878400, 23520, 1;
1, 241920, 948326400, 59744563200, 59744563200, 948326400, 241920, 1;
MATHEMATICA
(* First program *)
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[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 20 2021 *)
(* Second program *)
f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
T[n_, k_, m_]:= If[n==0, 1, f[n, m]/(f[k, m]*f[n-k, m])];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 20 2021 *)
PROG
(Sage)
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) )
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, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 20 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 15 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 20 2021
STATUS
approved