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

Author

Roger L. Bagula, Feb 23 2009

Keywords

• nonn
• tabl

Examples

			  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;
		

Links

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

Crossrefs

Cf. A008292, A157114, A157118.

Programs

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

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 *)

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 (kA157117(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

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).

Extensions

Edited by G. C. Greubel, Jan 11 2022