A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k).

0, 0, 0, 0, 2, 0, 0, 18, 6, 0, 0, 192, 72, 48, 0, 0, 2500, 960, 720, 540, 0, 0, 38880, 15000, 11520, 9720, 7680, 0, 0, 705894, 272160, 210000, 181440, 161280, 131250, 0, 0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0

Offset

0,5

Comments

A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.

Links

Table of n, a(n) for n=0..44.

John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.

Example

Triangle starts:
[0] [0]
[1] [0,        0]
[2] [0,        2,       0]
[3] [0,       18,       6,       0]
[4] [0,      192,      72,      48,      0]
[5] [0,     2500,     960,     720,     540,       0]
[6] [0,    38880,   15000,   11520,    9720,    7680,       0]
[7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
[8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]

Mathematica

A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)

Prog

(SageMath)
def T(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])

Crossrefs

T(n, 1) = A066274(n) for n >= 1.

T(n, 1)/(n - 1) = A000169(n) for n >= 2.

T(n, n - 1) = 2*A081133(n) for n >= 1.

Sum_{k=0..n} T(n, k) = A001864(n).

(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.

(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.

(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.

T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.

Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).

Sequence in context: A108474 A120582 A003784 * A244143 A066294 A363073

Adjacent sequences: A368846 A368847 A368848 * A368850 A368851 A368852

Keyword

nonn,tabl

Author

Peter Luschny, Jan 11 2024

Status

approved