A369016 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).

0, 0, 0, 0, 1, 0, 0, 6, 2, 0, 0, 48, 18, 12, 0, 0, 500, 192, 144, 108, 0, 0, 6480, 2500, 1920, 1620, 1280, 0, 0, 100842, 38880, 30000, 25920, 23040, 18750, 0, 0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0

Offset

0,8

Links

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

Formula

T = B066320 - A369017 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).

Example

Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,      0]
  [3] [0,       6,      2,      0]
  [4] [0,      48,     18,     12,      0]
  [5] [0,     500,    192,    144,    108,      0]
  [6] [0,    6480,   2500,   1920,   1620,   1280,      0]
  [7] [0,  100842,  38880,  30000,  25920,  23040,  18750,      0]
  [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]

Maple

T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):
seq(seq(T(n, k), k = 0..n), n=0..9);

Mathematica

A369016[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k-1); Table[A369016[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)

Prog

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

Crossrefs

A368849, A368982 and this sequence are alternative sum representation for A001864 with different normalizations.

T(n, k) = A368849(n, k) / n for n >= 1.

T(n, 1) = A053506(n) for n >= 1.

T(n, n - 1) = A055897(n - 1) for n >= 2.

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

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

Cf. A066320, A369017.

Sequence in context: A224476 A123147 A119831 * A224842 A318138 A130143

Adjacent sequences: A369013 A369014 A369015 * A369017 A369018 A369019

Keyword

nonn,tabl

Author

Peter Luschny, Jan 12 2024

Status

approved