A141686 Triangle read by rows: T(n, k) = binomial(n-1, k-1)*A008292(n, k).

1, 1, 1, 1, 8, 1, 1, 33, 33, 1, 1, 104, 396, 104, 1, 1, 285, 3020, 3020, 285, 1, 1, 720, 17865, 48320, 17865, 720, 1, 1, 1729, 90153, 546665, 546665, 90153, 1729, 1, 1, 4016, 409024, 4941104, 10933300, 4941104, 409024, 4016, 1, 1, 9117, 1722240, 38236128, 165104604, 165104604, 38236128, 1722240, 9117, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 08 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    8,       1;
  1,   33,      33,        1;
  1,  104,     396,      104,         1;
  1,  285,    3020,     3020,       285,         1;
  1,  720,   17865,    48320,     17865,       720,        1;
  1, 1729,   90153,   546665,    546665,     90153,     1729,       1;
  1, 4016,  409024,  4941104,  10933300,   4941104,   409024,    4016,    1;
  1, 9117, 1722240, 38236128, 165104604, 165104604, 38236128, 1722240, 9117, 1;
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of table, flattened

Crossrefs

Cf. A007318, A008292, A104098.

Programs

Haskell

a141686 n k = a141686_tabl !! (n-1) !! (k-1)
a141686_row n = a141686_tabl !! (n-1)
a141686_tabl = zipWith (zipWith (*)) a007318_tabl a008292_tabl
-- Reinhard Zumkeller, Apr 16 2014

Magma

A141686:= func< n, k | Binomial(n-1, k-1)*EulerianNumber(n, k-1) >;
[A141686(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 31 2024

Mathematica

(* Recurrence for A008292 *)
f[n_, k_]:= If[k==1||k==n,1, (n-k+1)*f[n-1,k-1] + k*f[n-1,k]];
Table[f[n, k]*Binomial[n-1,k-1], {n,12}, {k,n}]//Flatten
(* Second program *)
Needs["Combinatorica`"];
A141686[n_, k_]:= Binomial[n-1,k-1]*Eulerian[n,k-1];
Table[A141686[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Dec 31 2024 *)

Python

# or SageMath
from sage.combinat.combinat import eulerian_number
def A141686(n,k): return binomial(n-1,k-1)*eulerian_number(n,k-1)
print(flatten([[A141686(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Dec 31 2024

Formula

T(n, k) = T(n, n-k+1).

Sum_{k=1..n} T(n, k) = A104098(n) (row sums).

Extensions

keyword:tabl inserted, indices corrected by the Assoc. Eds. of the OEIS, Jun 30 2010