A369016 - OEIS

%I #22 Jan 28 2024 18:07:41

%S 0,0,0,0,1,0,0,6,2,0,0,48,18,12,0,0,500,192,144,108,0,0,6480,2500,

%T 1920,1620,1280,0,0,100842,38880,30000,25920,23040,18750,0,0,1835008,

%U 705894,544320,472500,430080,393750,326592,0

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

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

%e Triangle starts:

%e   [0] [0]

%e   [1] [0,       0]

%e   [2] [0,       1,      0]

%e   [3] [0,       6,      2,      0]

%e   [4] [0,      48,     18,     12,      0]

%e   [5] [0,     500,    192,    144,    108,      0]

%e   [6] [0,    6480,   2500,   1920,   1620,   1280,      0]

%e   [7] [0,  100842,  38880,  30000,  25920,  23040,  18750,      0]

%e   [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]

%p T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):

%p seq(seq(T(n, k), k = 0..n), n=0..9);

%t 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 *)

%o (SageMath)

%o def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1)

%o for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

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

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

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

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

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

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

%Y Cf. A066320, A369017.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Jan 12 2024