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)!.

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

Author

Peter Luschny, Nov 20 2020

Keywords

• nonn
• tabl

Comments

Related to the multinomial that is called M2 in Abramowitz and Stegun, p. 831.

Examples

			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]
		

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.

Crossrefs

Cf. A339034 (row sums), A092271.

Programs

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 *)

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))

Formula

T(n, k) = n! / A092271(n, k) for k > 0.