A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.

1, 1, 1, 2, 3, 2, 6, 11, 12, 5, 24, 50, 70, 50, 15, 120, 274, 450, 425, 225, 52, 720, 1764, 3248, 3675, 2625, 1092, 203, 5040, 13068, 26264, 33845, 29400, 16744, 5684, 877, 40320, 109584, 236248, 336420, 336735, 235872, 110838, 31572, 4140

Offset: 0

Author

Peter Luschny and Mélika Tebni, Jul 05 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) begins:
[0]    1;
[1]    1,      1;
[2]    2,      3,     2;
[3]    6,     11,    12,      5;
[4]   24,     50,    70,     50,    15;
[5]  120,    274,   450,    425,   225,     52;
[6]  720,   1764,  3248,   3675,  2625,   1092,  203;
[7] 5040,  13068, 26264,  33845, 29400,  16744, 5684, 877;
		

Crossrefs

Cf. A002720 (row sums), A000166 (alternating row sums), A000110 (main diagonal), A000142 (column 0).

Cf. A105479, A000254, A130534, A053557, A053556.

Programs

Maple

T := (n, k) -> (-1)^(n-k)*combinat:-bell(k)*Stirling1(n+1, k+1):
seq(seq(T(n, k), k = 0..n), n = 0..8);

Python

from functools import cache
@cache
def b(n: int, k=0):
    return int(n < 1) or k * b(n - 1, k) + b(n - 1, k + 1)
@cache
def s(n: int) -> list[int]:
    if n == 0: return [1]
    row = [0] + s(n - 1)
    for k in range(1, n): row[k] = row[k] + (n - 1) * row[k + 1]
    return row
def A355266_row(n):
    return [s * b(k - 1) for k, s in enumerate(s(n + 1))][1:]
for n in range(9): print(A355266_row(n))

Formula

T(n, k) = A000110(k) * A130534(n, k).

Sum_{k=0..n} T(n, k) = n!*Laguerre(n, -1) = A002720(n).

Sum_{k=0..n} (-1)^k*T(n, k) = !n = n!*A053557(n)/A053556(n) = A000166(n).