A157117 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k), read by rows.

2, 1, 1, 1, 8, 1, 1, 131, 131, 1, 1, 8204, 29216, 8204, 1, 1, 2097187, 44136233, 44136233, 2097187, 1, 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1, 1, 8796093022411, 150092195453359483, 288159195861579519, 288159195861579519, 150092195453359483, 8796093022411, 1

Offset

0,1

Links

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Formula

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k).

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

Example

  2;
  1,          1;
  1,          8,            1;
  1,        131,          131,            1;
  1,       8204,        29216,         8204,            1;
  1,    2097187,     44136233,     44136233,      2097187,          1;
  1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1;

Mathematica

f[n_, k_]:= If[k<=n, Eulerian[n*k+1, n-k], Eulerian[n*(n-k)+1, k]];
T[n_, k_]:= f[n, k] + f[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

Prog

(Magma)
Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
f:= func< n, k | k le n select Eulerian(n*k+1, n-k) else Eulerian(n*(n-k)+1, k) >;
A157117:= func< n, k | f(n, k) + f(n, n-k) >;
[A157117(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
(Sage)
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
def f(n, k): return Eulerian(n*k+1, n-k) if (k<n+1) else Eulerian(n*(n-k)+1, k)
def A157117(n, k): return f(n, k) + f(n, n-k)
flatten([[A157117(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

Crossrefs

Cf. A008292, A157114, A157118.

Sequence in context: A329043 A329042 A171660 * A322143 A264081 A061538

Adjacent sequences: A157114 A157115 A157116 * A157118 A157119 A157120

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 23 2009

Extensions

Edited by G. C. Greubel, Jan 11 2022

Status

approved