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A339033
Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!.
5
1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 3, 4, 24, 0, 5, 4, 6, 12, 120, 0, 6, 5, 8, 18, 48, 720, 0, 7, 6, 10, 24, 72, 240, 5040, 0, 8, 7, 12, 30, 96, 360, 1440, 40320, 0, 9, 8, 14, 36, 120, 480, 2160, 10080, 362880, 0, 10, 9, 16, 42, 144, 600, 2880, 15120, 80640, 3628800
OFFSET
0,5
COMMENTS
Related to the multinomial that is called M2 in Abramowitz and Stegun, p. 831.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, page 831.
FORMULA
T(n, k) = n! / A092271(n, k) for k > 0.
EXAMPLE
Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 2]
3: [0, 3, 2, 6]
4: [0, 4, 3, 4, 24]
5: [0, 5, 4, 6, 12, 120]
6: [0, 6, 5, 8, 18, 48, 720]
7: [0, 7, 6, 10, 24, 72, 240, 5040]
8: [0, 8, 7, 12, 30, 96, 360, 1440, 40320]
9: [0, 9, 8, 14, 36, 120, 480, 2160, 10080, 362880]
MATHEMATICA
A339033[n_, k_] := Which[k == 0, Boole[n == 0], n == k, n!, True, (n+1-k)*(k-1)!];
Table[A339033[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 31 2024 *)
PROG
(SageMath)
def A339033(n, k):
if k == 0: return 0^n
if n == k: return factorial(n)
return (n + 1 - k)*factorial(k - 1)
for n in (0..10): print([A339033(n, k) for k in (0..n)])
def A339033Row(n):
S = [0^n]
for k in range(n, 0, -1):
for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
S.append(p.aut())
return S
for n in (0..10): print(A339033Row(n))
CROSSREFS
Cf. A339034 (row sums), A092271.
Sequence in context: A108807 A129236 A127465 * A327028 A271707 A341445
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 20 2020
STATUS
approved