A152684
a(n) is the number of top-down sequences (F_1, F_2, ..., F_n) whereas each F_i is a labeled forest on n nodes, containing i directed rooted trees. F_(i+1) is proper subset of F_i.
Original entry on oeis.org
1, 2, 18, 384, 15000, 933120, 84707280, 10569646080, 1735643790720, 362880000000000, 94121726392108800, 29658516531078758400, 11159820050604594969600, 4942478402320838374195200, 2544989406021562500000000000, 1507645899890367707813511168000
Offset: 1
a(1) = 1^(1-2)*(1!) = 1.
a(2) = 2^(2-2)*(2!) = 2.
a(3) = 3^(3-2)*(3!) = 18.
- Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill 2007, Page 276.
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[Factorial(n-1)*n^(n-1): n in [1..20]]; // G. C. Greubel, Nov 28 2022
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a:= proc(n) option remember; `if`(n=1, 1,
a(n-1)*(n/(n-1))^(n-3)*n^2)
end:
seq(a(n), n=1..20); # Alois P. Heinz, May 16 2013
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Table[n^(n - 1) (n - 1)!, {n, 1, 16}] (* Geoffrey Critzer, May 10 2013 *)
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[factorial(n-1)*n^(n-1) for n in range(1,21)] # G. C. Greubel, Nov 28 2022
A137268
Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.
Original entry on oeis.org
1, 2, 2, 4, 6, 6, 8, 18, 24, 24, 16, 54, 96, 120, 120, 32, 162, 384, 600, 720, 720, 64, 486, 1536, 3000, 4320, 5040, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880
Offset: 1
Triangle begins as:
1;
2, 2;
4, 6, 6;
8, 18, 24, 24;
16, 54, 96, 120, 120;
32, 162, 384, 600, 720, 720;
64, 486, 1536, 3000, 4320, 5040, 5040;
128, 1458, 6144, 15000, 25920, 35280, 40320, 40320;
256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880;
512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;
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[Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 28 2022
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T[n_, k_]:= k!*(k+1)^(n-k);
Table[T[n, k], {n, 12}, {k, n}]//Flatten
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def A137268(n,k): return factorial(k)*(k+1)^(n-k)
flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # G. C. Greubel, Nov 28 2022
A342811
Volume of the permutohedron obtained from the coordinates 1, 2, 4, ..., 2^(n-1), multiplied by (n-1)!.
Original entry on oeis.org
1, 13, 1009, 354161, 496376001, 2632501072321, 52080136110870785, 3872046158193220660993, 1099175272489026844687825921, 1210008580962784935280673680079873, 5225407816779297641534116390319222362113
Offset: 2
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a[n_] := Sum[(p.(2^Range[0, n-1]))^(n-1) / Times @@ Differences[p], {p, Permutations@Range@n}];
Table[a[n], {n, 2, 8}]
Showing 1-3 of 3 results.
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