A369016 -id:A369016 - cp's OEIS frontend

A368982 Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.

0, 0, 0, 0, 1, 0, 0, 9, 3, 0, 0, 96, 36, 24, 0, 0, 1250, 480, 360, 270, 0, 0, 19440, 7500, 5760, 4860, 3840, 0, 0, 352947, 136080, 105000, 90720, 80640, 65625, 0, 0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0

Offset: 0

Peter Luschny, Jan 11 2024

			Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,       0]
  [3] [0,       9,       3,       0]
  [4] [0,      96,      36,      24,       0]
  [5] [0,    1250,     480,     360,     270,       0]
  [6] [0,   19440,    7500,    5760,    4860,    3840,       0]
  [7] [0,  352947,  136080,  105000,   90720,   80640,   65625,       0]
  [8] [0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0]

A368849, A369016 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / 2.
T(n, 1) = A081131(n) for n >= 1.
T(n, n - 1) = A081133(n - 2) for n >= 2.
Sum_{k=0..n} T(n, k) = A036276(n - 1) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / 2 for n >= 0.
Cf. A369072, A369025.

T := (n, k) -> binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)/2:
seq(seq(T(n, k), k = 0..n), n=0..9);

A368982[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k)/2; Table[A368982[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)

def T(n, k): return binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)//2
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

T = A369072 - A369025.

A369017 Triangle read by rows: T(n, k) = binomial(n-1, k-1) * (k - 1)^(k - 1) * k * (n - k + 1)^(n - k - 1).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 3, 4, 12, 0, 16, 18, 36, 108, 0, 125, 128, 216, 432, 1280, 0, 1296, 1250, 1920, 3240, 6400, 18750, 0, 16807, 15552, 22500, 34560, 57600, 112500, 326592, 0, 262144, 235298, 326592, 472500, 716800, 1181250, 2286144, 6588344

Offset: 0

Peter Luschny, Jan 12 2024

			Triangle starts:
[0][0]
[1][0,      1]
[2][0,      1,      2]
[3][0,      3,      4,     12]
[4][0,     16,     18,     36,    108]
[5][0,    125,    128,    216,    432,   1280]
[6][0,   1296,   1250,   1920,   3240,   6400,   18750]
[7][0,  16807,  15552,  22500,  34560,  57600,  112500,  326592]
[8][0, 262144, 235298, 326592, 472500, 716800, 1181250, 2286144, 6588344]

Winston de Greef, Table of n, a(n) for the first 150 rows, flattened (n = 0..11324)

Cf. A066320, A369016.

T(n, k) = binomial(n-1, k-1)*(k-1)^(k-1)*k*(n-k+1)^(n-k-1)
for n in 0:9 (println([T(n, k) for k in 0:n])) end

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

A369017[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] k (n-k+1)^(n-k-1);
Table[A369017[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)

T(n, k) = binomial(n-1, k-1) * (k - 1)^(k - 1) * k * (n - k + 1)^(n - k - 1) \\ Winston de Greef, Jan 27 2024

T = B066320 - A369016 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).

cp's OEIS Frontend

A368982 Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.

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