A157221 Triangle T(n, k) = Eulerian(n*f(n, k) + 1, f(n, k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 14608, 26, 1, 1, 57, 152637, 152637, 57, 1, 1, 120, 1479726, 251732291184, 1479726, 120, 1, 1, 247, 13824739, 16871482830550, 16871482830550, 13824739, 247, 1, 1, 502, 126781020, 1103881308184906, 113909683214485984529600, 1103881308184906, 126781020, 502, 1

Offset

0,5

Links

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

Formula

T(n, k) = Eulerian(n*f(n, k) + 1, f(n, k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k.

Example

Triangle begins as:
  1;
  1,   1;
  1,   4,        1;
  1,  11,       11,              1;
  1,  26,    14608,             26,              1;
  1,  57,   152637,         152637,             57,        1;
  1, 120,  1479726,   251732291184,        1479726,      120,   1;
  1, 247, 13824739, 16871482830550, 16871482830550, 13824739, 247, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k+1-j)^n, {j, 0, k+1}];
T[n_, k_]:= Eulerian[n*f[n, k] + 1, f[n, k]];
Table[Eulerian[n*f[n, k] +1, f[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n*f(n, k)+1, f(n, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2022
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
flatten([[Eulerian(n*f(n, k)+1, f(n, k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A008292, A157219.

Sequence in context: A174526 A008292 A174036 * A146967 A173152 A156049

Adjacent sequences: A157218 A157219 A157220 * A157222 A157223 A157224

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved