OFFSET
0,6
FORMULA
T(n, k) = k*binomial(n, k - 1) for n >= 1 and 0 < k < n, and T(n, 0) = 0^n, T(n, n) = n.
EXAMPLE
Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1, 2]
3: [0, 1, 6, 3]
4: [0, 1, 8, 18, 4]
5: [0, 1, 10, 30, 40, 5]
6: [0, 1, 12, 45, 80, 75, 6]
7: [0, 1, 14, 63, 140, 175, 126, 7]
8: [0, 1, 16, 84, 224, 350, 336, 196, 8]
9: [0, 1, 18, 108, 336, 630, 756, 588, 288, 9]
.
T(4, 1) = 1 = 1*card({1234})
T(4, 2) = 8 = 2*card({123|4, 124|3, 134|2, 1|234})
T(4, 3) = 18 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
T(4, 4) = 4 = 4*card({1|2|3|4})
MAPLE
PROG
(SageMath) # Shows the combinatorial interpretation.
def A339031Row(n):
if n == 0: return [1]
M = matrix(n + 2)
for k in (1..n):
for p in SetPartitions(n):
if p.max_block_size() == k:
M[len(p), k] += p.cardinality()
return [M[k, n-k+1] for k in (0..n)]
for n in (0..9): print(A339031Row(n))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 22 2020
STATUS
approved