A118537 -id:A118537 - cp's OEIS frontend

A055897 a(n) = n*(n-1)^(n-1).

1, 2, 12, 108, 1280, 18750, 326592, 6588344, 150994944, 3874204890, 110000000000, 3423740047332, 115909305827328, 4240251492291542, 166680102383370240, 7006302246093750000, 313594649253062377472, 14890324713954061755186, 747581753430634213933056

Offset: 1

Christian G. Bower, Jun 12 2000

nonn

Total number of leaves in all labeled rooted trees with n nodes.
Number of endofunctions of [n] such that no element of [n-1] is fixed. E.g., a(3)=12: 123 -> 331, 332, 333, 311, 312, 313, 231, 232, 233, 211, 212, 213.
Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n). - Warut Roonguthai, May 06 2006
Determinant of the n X n matrix ((2n, n^2, 0, ..., 0), (1, 2n, n^2, 0, ..., 0), (0, 1, 2n, n^2, 0, ..., 0), ..., (0, ..., 0, 1, 2n)). - Michel Lagneau, May 04 2010
For n > 1: a(n) = A240993(n-1) / A240993(n-2). - Reinhard Zumkeller, Aug 31 2014
Total number of points m such that f^(-1)(m) = {m}, (i.e., the preimage of m is the singleton set {m}) summed over all functions f:[n]->[n]. - Geoffrey Critzer, Jan 20 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Frank Ellermann, Illustration of binomial transforms
Index entries for sequences related to rooted trees

Cf. A003227, A003228, A055314, A055540, A055541, A060226, A118537.

Cf. A240993. Diagonal of A265583.

List([1..20], n-> n*(n-1)^(n-1)); # G. C. Greubel, Aug 10 2019

a055897 n = n * (n - 1) ^ (n - 1) -- Reinhard Zumkeller, Aug 31 2014

[n*(n-1)^(n-1): n in [1..20]] // Wesley Ivan Hurt, Jun 26 2014

A055897:=n->`if`(n=1,1,n*(n-1)^(n-1)); seq(A055897(n), n=1..20); # Wesley Ivan Hurt, Jun 26 2014

Join[{1},Table[n(n-1)^(n-1), {n,2,20}]] (* Harvey P. Dale, Jul 18 2011 *)

{a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ Paul D. Hanna, Dec 27 2012

[n*(n-1)^(n-1) for n in (1..20)] # G. C. Greubel, Aug 10 2019

E.g.f.: x/(1-T), where T=T(x) is Euler's tree function (see A000169).
a(n) = Sum_{k=1..n} A055302(n, k)*k.
a(n) = the n-th term of the (n-1)-th binomial transform of {1, 1, 4, 18, 96, ..., (n-1)*(n-1)!, ...} (cf. A001563). - Paul D. Hanna, Nov 17 2003
a(n) = (n-1)^(n-1) + Sum_{i=2..n} (n-1)^(n-i)*binomial(n-1, i-1)*(i-1) *(i-1)!. - Paul D. Hanna, Nov 17 2003
a(n) = [x^(n-1)] 1/(1 - (n-1)*x)^2. - Paul D. Hanna, Dec 27 2012
a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Nov 14 2014

Additional comments from Vladeta Jovovic, Mar 31 2001 and Len Smiley, Dec 11 2001

A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 12, 6, 2, 0, 0, 20, 24, 18, 0, 0, 0, 30, 60, 84, 30, 2, 0, 0, 42, 120, 260, 240, 66, 0, 0, 0, 56, 210, 630, 1020, 732, 126, 2, 0, 0, 72, 336, 1302, 3120, 4100, 2184, 258, 0, 0, 0, 90, 504, 2408, 7770, 15630, 16380, 6564, 510, 2, 0, 0, 110

Offset: 1

Joshua Zucker, May 29 2005

Note that we keep one edge in the circular graph even when there's only one node (so there are 0 colorings of one node with k colors).

Number of closed walks of length n on the complete graph K_{k}. - Andrew Howroyd, Mar 12 2017

			From _Andrew Howroyd_, Mar 12 2017: (Start)
Table begins:
  0 0   0     0      0       0        0        0         0 ...
  0 2   6    12     20      30       42       56        72 ...
  0 0   6    24     60     120      210      336       504 ...
  0 2  18    84    260     630     1302     2408      4104 ...
  0 0  30   240   1020    3120     7770    16800     32760 ...
  0 2  66   732   4100   15630    46662   117656    262152 ...
  0 0 126  2184  16380   78120   279930   823536   2097144 ...
  0 2 258  6564  65540  390630  1679622  5764808  16777224 ...
  0 0 510 19680 262140 1953120 10077690 40353600 134217720 ...
(End)
a(4,3) = 18 because there are three choices for the first node's color (call it 1) and then two choices for the second node's color (call it 2) and then the remaining two nodes can be 12, 13, or 32. So in total there are 3*2*3 = 18 ways. a(3,4) = 4*3*2 = 24 because the three nodes must be three distinct colors.

Andrew Howroyd, Table of n, a(n) for n = 1..1274

Columns include A092297, A226493. Main diagonal is A118537.
Rows 2-7 are A002378, A007531, A091940, A061167, A131472, A133499.
Cf. A090860, A208535.

a(n, k) = (k-1)^n + (-1)^n * (k-1).

a(67) corrected by Andrew Howroyd, Mar 12 2017

A172286 Numbers of circuits of length 2n in K_{n,n} (the complete bipartite graph on 2n vertices).

Original entry on oeis.org

2, 32, 1458, 131072, 19531250, 4353564672, 1356446145698, 562949953421312, 300189270593998242, 200000000000000000000, 162805498773679522226642, 158993694406781688266883072, 183466660386537233316799232018

Offset: 1

Thibaut Lienart (syncthib(AT)gmail.com), Jan 30 2010

Circuits are allowed to be self-intersecting and are directional with a designated start node. The number of (self-avoiding) directed cycles is given by A010790. - Andrew Howroyd, Sep 05 2018

			a(2) = 32 because there are 32 circuits of length 4 in the complete bipartite graph K2,2.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A010790, A118537.

nmax = 10;
for k=1:nmax
an = 2*k^(2*k);
fprintf('%3.0f ', an);
end

a(n)=2*n^(2*n); \\ Andrew Howroyd, Sep 05 2018

a(n) = 2*n^(2*n).

More terms from Max Alekseyev, Jan 18 2012

cp's OEIS Frontend

A055897 a(n) = n*(n-1)^(n-1).

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A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).

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Formula

Extensions

A172286 Numbers of circuits of length 2n in K_{n,n} (the complete bipartite graph on 2n vertices).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

PARI

Formula

Extensions