A385895 Table read by rows: T(n, k) = T(n, k-1) + m * T(n-1, n-k) for k > 1, T(n, 1) = T(n-1, n-1), and T(n, 0) = 0^n, for m = 2.

1, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 3, 9, 11, 11, 0, 11, 33, 51, 57, 57, 0, 57, 171, 273, 339, 361, 361, 0, 361, 1083, 1761, 2307, 2649, 2763, 2763, 0, 2763, 8289, 13587, 18201, 21723, 23889, 24611, 24611, 0, 24611, 73833, 121611, 165057, 201459, 228633, 245211, 250737, 250737

Offset: 0

Author

Peter Luschny, Jul 20 2025

Keywords

• nonn
• tabl

Comments

The sequence extends the generalized Euler numbers A001586 to a regular table by a parametrized Seidel transformation (see the Python program) that for the case m = 1 leads to the Euler-Bernoulli numbers A008281.

Examples

			Triangle begins:
  [0] 1;
  [1] 0,    1;
  [2] 0,    1,    1;
  [3] 0,    1,    3,     3;
  [4] 0,    3,    9,    11,    11;
  [5] 0,   11,   33,    51,    57,    57;
  [6] 0,   57,  171,   273,   339,   361,   361;
  [7] 0,  361, 1083,  1761,  2307,  2649,  2763,  2763;
  [8] 0, 2763, 8289, 13587, 18201, 21723, 23889, 24611, 24611;
		

Crossrefs

Cf. A001586 (main diagonal), A123110 (m=0), A008281 (m=1), this sequence (m=2).

Programs

Maple

T := proc(n, k) option remember; ifelse(k = 0, 0^n, ifelse(k = 1, T(n-1, n-1), T(n, k-1) + 2*T(n-1, n-k))) end: seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_] := T[n, k] =
  Which[
    k == 0, Boole[n == 0],
    k == 1, T[n - 1, n - 1],
    True, T[n, k - 1] + 2*T[n - 1, n - k]
  ];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Python

from functools import cache
@cache
def seidel(n: int, m: int) -> list[int]:
    if n == 0: return [1]
    rowA = seidel(n - 1, m)
    row = [0] + rowA
    row[1] = row[n]
    for k in range(2, n + 1):
        row[k] = row[k - 1] + m * rowA[n - k]
    return row
def A385895row(n: int) -> list[int]: return seidel(n, 2)
for n in range(9): print(A385895row(n))