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).
Original entry on oeis.org
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
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]
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, 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.
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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);
-
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 *)
-
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)])
A066320
Triangle read by rows: T(n, k) = binomial(n, k)*k^k*(n-k)^(n-k-1) k=0..n-1.
Original entry on oeis.org
1, 2, 2, 9, 6, 12, 64, 36, 48, 108, 625, 320, 360, 540, 1280, 7776, 3750, 3840, 4860, 7680, 18750, 117649, 54432, 52500, 60480, 80640, 131250, 326592, 2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344, 43046721
Offset: 1
Triangle starts:
[1][ 1]
[2][ 2, 2]
[3][ 9, 6, 12]
[4][ 64, 36, 48, 108]
[5][ 625, 320, 360, 540, 1280]
[6][ 7776, 3750, 3840, 4860, 7680, 18750]
[7][ 117649, 54432, 52500, 60480, 80640, 131250, 326592]
[8][2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344]
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 68 (2.1.43).
T = n *
A185390 after proper alignment of offsets.
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# Assuming offset (n=1, k=1).
T(n, k) = binomial(n-1, k-1)*(k-1)^(k-1)*n*(n-k+1)^(n-k-1)
for n in 1:9 (println([n], [T(n, k) for k in 1:n])) end
# Peter Luschny, Jan 12 2024
A350297
Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.
Original entry on oeis.org
1, 1, 0, 2, 2, 1, 6, 12, 12, 8, 24, 72, 108, 108, 81, 120, 480, 960, 1280, 1280, 1024, 720, 3600, 9000, 15000, 18750, 18750, 15625, 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936, 40320, 282240, 987840, 2304960, 4033680, 5647152, 6588344, 6588344, 5764801
Offset: 0
Triangle T(n,k) begins:
-----------------------------------------------------------------
n\k 0 1 2 3 4 5 6 7
-----------------------------------------------------------------
0 | 1,
1 | 1, 0,
2 | 2, 2, 1,
3 | 6, 12, 12, 8,
4 | 24, 72, 108, 108, 81,
5 | 120, 480, 960, 1280, 1280, 1024,
6 | 720, 3600, 9000, 15000, 18750, 18750, 15625,
7 | 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936.
...
-
T := (n, k) -> (n!/k!)*(n - 1)^k:
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Dec 24 2021
-
T[1, 0] := 1; T[n_, k_] := n!*(n - 1)^k/k!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 24 2021 *)
A225723
Triangular array read by rows: T(n,k) is the number of size k components in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n}; n>=1, 1<=k<=n.
Original entry on oeis.org
1, 2, 3, 12, 9, 17, 108, 72, 68, 142, 1280, 810, 680, 710, 1569, 18750, 11520, 9180, 8520, 9414, 21576, 326592, 196875, 152320, 134190, 131796, 151032, 355081, 6588344, 3919104, 2975000, 2544640, 2372328, 2416512, 2840648, 6805296
Offset: 1
Triangle T(n,k) begins:
1;
2, 3;
12, 9, 17;
108, 72, 68, 142;
1280, 810, 680, 710, 1569;
18750, 11520, 9180, 8520, 9414, 21576;
326592, 196875, 152320, 134190, 131796, 151032, 355081;
...
-
b:= n-> n!*add(n^(n-k-1)/(n-k)!, k=1..n):
T:= (n, k)-> binomial(n,k)*b(k)*(n-k)^(n-k):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, May 13 2013
-
nn = 8; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy =
Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
Drop[Range[0, nn]! CoefficientList[
Series[Log[1/(1 - txy)]/(1 - tx), {x, 0, nn}], {x, y}],
1]] // Grid
A322406
a(n) = n + n*n^n.
Original entry on oeis.org
2, 10, 84, 1028, 15630, 279942, 5764808, 134217736, 3486784410, 100000000010, 3138428376732, 106993205379084, 3937376385699302, 155568095557812238, 6568408355712890640, 295147905179352825872, 14063084452067724991026, 708235345355337676357650, 37589973457545958193355620
Offset: 1
a(3) = 3 + 3*3^3 = 3 + 3*27 = 8 + 81 = 84.
A353122
Numbers k such that k^k*(k+1) + 1 is prime.
Original entry on oeis.org
0, 1, 2, 3, 6, 9, 186, 198, 8390
Offset: 1
9 is in the sequence because 9^9*(9+1) + 1 = 3874204891, which is prime.
-
[n: n in [0..200] | IsPrime(n^n*(n+1) + 1)];
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Join[{0}, Select[Range[200], PrimeQ[#^#*(# + 1) + 1] &]] (* Amiram Eldar, Apr 25 2022 *)
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isok(k) = ispseudoprime(k^k*(k+1) + 1); \\ Michel Marcus, May 16 2022
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