A256550
Triangle read by rows, T(n,k) = EL(n,k)/(n-k+1)! and EL(n,k) the matrix-exponential of the unsigned Lah numbers scaled by exp(-1), for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 12, 6, 1, 0, 15, 50, 40, 10, 1, 0, 52, 225, 250, 100, 15, 1, 0, 203, 1092, 1575, 875, 210, 21, 1, 0, 877, 5684, 10192, 7350, 2450, 392, 28, 1, 0, 4140, 31572, 68208, 61152, 26460, 5880, 672, 36, 1
Offset: 0
Triangle starts:
1;
0, 1;
0, 1, 1;
0, 2, 3, 1;
0, 5, 12, 6, 1;
0, 15, 50, 40, 10, 1;
0, 52, 225, 250, 100, 15, 1;
0, 203, 1092, 1575, 875, 210, 21, 1;
-
def T(dim) :
M = matrix(ZZ, dim)
for n in range(dim) :
M[n, n] = 1
for k in range(n) :
M[n,k] = (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
E = M.exp()/exp(1)
for n in range(dim) :
for k in range(n) :
M[n,k] = E[n,k]/factorial(n-k+1)
return M
T(8) # Computes the sequence as a lower triangular matrix.
A355266
Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.
Original entry on oeis.org
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
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;
-
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);
-
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))