A156767 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 = 3, read by rows.

1, 1, 1, 1, 10, 1, 1, 126, 126, 1, 1, 2040, 25704, 2040, 1, 1, 40920, 8347680, 8347680, 40920, 1, 1, 982800, 4021617600, 65111904000, 4021617600, 982800, 1, 1, 27523440, 2705003683200, 878482148544000, 878482148544000, 2705003683200, 27523440, 1

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 = 3.

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 = 3. - G. C. Greubel, Jun 19 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,     10,          1;
  1,    126,        126,           1;
  1,   2040,      25704,        2040,          1;
  1,  40920,    8347680,     8347680,      40920,      1;
  1, 982800, 4021617600, 65111904000, 4021617600, 982800, 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, 3], {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, 3], {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, 3): 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, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

Crossrefs

Cf. A007318 (m=0), A156765 (m=1), A156766 (m=2), this sequence (m=3).

Sequence in context: A158117 A172378 A015124 * A365025 A174921 A010180

Adjacent sequences: A156764 A156765 A156766 * A156768 A156769 A156770

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 15 2009

Extensions

Edited by G. C. Greubel, Jun 19 2021

Status

approved