A123110 -id:A123110 - cp's OEIS frontend

A114607 Start with 1 0 1 0 then add a one every time (e.g. 1 1 0 1 1 1 0 1 1 1 1 0 ...).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Dan M. Smith (SmittyDa33(AT)yahoo.com), Dec 14 2005

nonn

Cf. A073424.

Essentially the same sequence as A023532 and A123110. - N. J. A. Sloane, Feb 07 2020

Flatten[Join[{1,0},Table[Join[PadRight[{},n,1],{0}],{n,20}]]] (* Harvey P. Dale, Dec 23 2012 *)

a(n)=A023532(n-2), n>1. [From R. J. Mathar, Aug 11 2008]

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.

Original entry on oeis.org

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

Peter Luschny, Jul 20 2025

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.

			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;

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

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);

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

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))

cp's OEIS Frontend

A114607 Start with 1 0 1 0 then add a one every time (e.g. 1 1 0 1 1 1 0 1 1 1 1 0 ...).

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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.

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Python