A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 60, 240, 60, 1, 1, 360, 7200, 7200, 360, 1, 1, 2520, 302400, 1512000, 302400, 2520, 1, 1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1, 1, 181440, 1219276800, 256048128000, 1536288768000, 256048128000, 1219276800, 181440, 1

Offset: 0

Author

Roger L. Bagula, Feb 10 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     3,        1;
  1,    12,       12,         1;
  1,    60,      240,        60,         1;
  1,   360,     7200,      7200,       360,        1;
  1,  2520,   302400,   1512000,    302400,     2520,     1;
  1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1;
		

Links

• G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A156764 (m=3).

Cf. A009963.

Programs

Maple

SF := n -> mul(j!, j=0..n): T := (n,k) -> SF(n-1)/(SF(n-k)*SF(k)):
seq(print(seq(T(n,k),k=1..n-1)),n=0..9); # Peter Luschny, Jan 24 2015

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, 1], {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,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 20 2021 *)

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,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 21 2021

Formula

From G. C. Greubel, Jun 21 2021: (Start)

T(n, k) = BarnesG(n+3)/(BarnesG(k+3)*BarnesG(n-k+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 = 1. (End)

Extensions

New name and editing, Peter Luschny, Jan 24 2015