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

0,5

Links

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

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)

Example

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;

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 *)

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

Crossrefs

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

Cf. A009963.

Sequence in context: A098778 A078122 A128592 * A209424 A129619 A094573

Adjacent sequences: A156581 A156582 A156583 * A156585 A156586 A156587

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 10 2009

Extensions

New name and editing, Peter Luschny, Jan 24 2015

Status

approved