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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, 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
0,4
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).
EXAMPLE
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;
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);
PROG
(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))
CROSSREFS
Cf. A002720 (row sums), A000166 (alternating row sums), A000110 (main diagonal), A000142 (column 0).
Sequence in context: A087454 A059446 A298854 * A188881 A143806 A276551
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny and Mélika Tebni, Jul 05 2022
STATUS
approved