A104098 -id:A104098 - cp's OEIS frontend

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

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

			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;

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

Cf. A007318, A008292, A104098.

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

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

(* 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 *)

# 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

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

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

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

A011818 Normalized volume of center slice of n-dimensional cube: 2^n* n!*Vol({ (x_1,...,x_n) in [ 0,1 ]^n: n/2 <= Sum_{i = 1..n} x_i <= (n+1)/2 }).

Original entry on oeis.org

1, 3, 16, 115, 1056, 11774, 154624, 2337507, 39984640, 763546234, 16101629952, 371644257582, 9319104528384, 252270887452380, 7332475985461248, 227761317947788323, 7529455986838732800, 263948439074152148450

Offset: 1

Guenter M. Ziegler (ziegler(AT)math.tu-berlin.de)

D. Chakerian, D. Logothetti, CubeSlices, Pictorial Triangles, and Probability, Math. Mag., Vol. 64 (1991) 219-241.

Cf. A008292, A025585, A104098.

a := n -> add(binomial(n,k)*eulerian1(n,k), k=0..n-1):
seq(a(n), n=1..18); # Peter Luschny, Jun 30 2016

Eulerian1[n_, k_] = Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}];
a[n_] := Sum[Binomial[n, k] Eulerian1[n, k], {k, 0, n-1}];
Array[a, 18] (* Jean-François Alcover, Jun 03 2019 *)

V(d) = sum_{k=1}^{d-1} {d choose k-1} A_{d, k} where A_{k, d} denotes the Eulerian number (permutations of a d-set with k-1 descents) - see A008292.
Restated: a(n) = Sum_{k = 1..n} C(n,k-1)*A008292(n,k) for n>=1.
From Peter Bala, Jun 28 2016: (Start)
a(n) = 1/2*Sum_{k = 0..floor((n+1)/2)} (-1)^k*binomial(n + 1,k)*(n + 1 - 2*k)^n.
a(n) ~ sqrt(3)/2*(2/e)^(n+1)*(n+1)^n. (End)
a(2*n-1)/2^(2*n-2) = A025585(n) for n>=1. - Peter Luschny, Jun 30 2016

More terms from Paul D. Hanna, Mar 15 2006

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Python

Formula

Extensions