A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

1, 1, 1, 1, 6, 8, 1, 21, 108, 162, 1, 60, 800, 3840, 6144, 1, 155, 4500, 48750, 225000, 375000, 1, 378, 21672, 453600, 4354560, 19595520, 33592320, 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720

Offset: 0

Author

Peter Luschny, Apr 26 2024

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  [0] 1;
  [1] 1,   1;
  [2] 1,   6,     8;
  [3] 1,  21,   108,     162;
  [4] 1,  60,   800,    3840,     6144;
  [5] 1, 155,  4500,   48750,   225000,    375000;
  [6] 1, 378, 21672,  453600,  4354560,  19595520,   33592320;
  [7] 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720;
		

Crossrefs

Cf. A061711 (main diagonal), A066524 (column 1), A372312 (row sums).

Cf. A163626, A173018 (eulerian1).

Programs

Maple

S := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
row := n -> local k; seq(S(n, k) * n^k, k = 0..n):
seq(row(n), n = 0..8);

SageMath

def A372311_row(n) :
    x = polygen(ZZ, 'x')
    A = []
    for m in range(0, n + 1, 1) :
        A.append((-x)^m)
        for j in range(m, 0, -1):
            A[j - 1] = j * (A[j - 1] - A[j])
    return [n^k*c for k, c in enumerate(A[0])]
for n in (0..7) : print(A372311_row(n))