A394582 Square array read by ascending antidiagonals: T(n, k) = n * T(n, k-1) + (k mod 2) * T(n-1, k), and T(n, 1) = T(1, k) = 1.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 14, 10, 1, 1, 5, 30, 42, 21, 1, 1, 6, 55, 120, 147, 42, 1, 1, 7, 91, 275, 627, 441, 85, 1, 1, 8, 140, 546, 2002, 2508, 1408, 170, 1, 1, 9, 204, 980, 5278, 10010, 11440, 4224, 341, 1, 1, 10, 285, 1632, 12138, 31668, 61490, 45760, 13013, 682, 1

Offset

1,5

Comments

This array is introduced as "Extended central factorial numbers of the second kind". It acts as a structured merger of two layers: the odd-indexed columns form the central factorial triangle A008957, while the even-indexed columns form its scaling counterpart, the triangle A394813.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See pp. 2-6, 14.

Formula

T(n, k) represents the normalized central values of the (2*n - 1)-th finite differences of the sequence f(j) = j^(2*n + k - 2). Odd-indexed columns T(n, 2*m - 1) represent columns of the tangent-secant triangle A036969. Thus T(n, k) is the (2*n - 1)-th finite difference of the sequence f(j) = j^m (where m = 2*n + k - 2), evaluated at the central offset j = -(n - 1), and divided by (2*n - 1)!.

T(n, k) = (1/(2*n-1)!) * Sum_{i=0..2*n-1} (-1)^(2*n-1-i) * binomial(2*n-1, i) * (i-n+1)^(2*n+k-2).

T(n, k) = Sum_{j=0..2*n+k-2} binomial(2*n+k-2, j) * (1-n)^(2*n+k-2-j) * Stirling2(j, 2*n-1).

T(n, 2*k-1) = A008957(k+n-1, n) for n, k >= 1.

T(n, 2*k) = n * A008957(k+n-1, n) for n, k >= 1.

T(n, 2*k) = A394813(k+n-1, k).

T(n, 2*k-1) = Sum_{i_{k-1}=1..n} Sum_{i_{k-2}=1..i_{k-1}} ... Sum_{i_1=1..i_2} (Product_{j=1..k-1} i_j)^2 for n, k >= 1.

T(n, 2*k) = n * Sum_{i_{k-1}=1..n} Sum_{i_{k-2}=1..i_{k-1}} ... Sum_{i_1=1..i_2} (Product_{j=1..k-1} i_j)^2 for n, k >= 1.

T(n, k) = A395021(2*n-2+k, 2*n-1).

Example

The square array T(n, k) begins:
  n=1:  1,  1,   1,    1,    1,     1,      1,       1,        1,        1
  n=2:  1,  2,   5,   10,   21,    42,     85,     170,      341,      682
  n=3:  1,  3,  14,   42,  147,   441,   1408,    4224,    13013,    39039
  n=4:  1,  4,  30,  120,  627,  2508,  11440,   45760,   196053,   784212
  n=5:  1,  5,  55,  275, 2002, 10010,  61490,  307450,  1733303,  8666515
  n=6:  1,  6,  91,  546, 5278, 31668, 251498, 1508988, 10787231, 64723386
  ...

Maple

T := proc(n, k) option remember; if n=1 or k = 1 then 1 else n * T(n, k-1) + irem(k, 2) * T(n-1, k) fi end: seq(print(seq(T(n, k), k = 1..10)), n = 1..8); # Peter Luschny, Apr 02 2026

Mathematica

A394582[n_, k_] := A394582[n, k] = If[n == 1 || k == 1, 1, n*A394582[n, k-1] + Mod[k, 2]*A394582[n-1, k]];
Table[A394582[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Apr 04 2026 *)

Prog

(Python)
from functools import cache
@cache
def T(n, k): # Recurrence based on parity of k
    if n == 1 or k == 1: return 1
    r = n * T(n, k - 1)
    return r + T(n - 1, k) if k % 2 else r
for n in range(1, 7): print([T(n, k) for k in range(1, 11)])
print([T(k-n+1, n) for k in range(1, 12) for n in range(1, k+1)])

Crossrefs

Cf. A008957, A036969, A002450, A002451, A000027 (column 2), A000330 (column 3), A108678 (column 4), A060493 (column 5), A394883 (column 6), A351105 (column 7), A394896 (column 8), A000012 (row 1), A000975 (row 2), A394882 (row 3), A394763 (main diagonal), A394813, A395021 (parent).

Sequence in context: A340968 A128198 A320031 * A123349 A123352 A398069

Adjacent sequences: A394579 A394580 A394581 * A394583 A394584 A394585

Keyword

tabl,nonn,easy

Author

Husiev Andrii Alekseevich, Mar 25 2026

Status

approved