A055540 -id:A055540 - 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

A055541 Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.

Original entry on oeis.org

0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658, 41532319635035234107392, 2082547005958224830656820

Offset: 1

Eric W. Weisstein

nonn

Equivalently, a(n) is the number of rooted labeled trees such that the root node has degree 1. - Geoffrey Critzer, Feb 07 2012

G. C. Greubel, Table of n, a(n) for n = 1..385
Dixy Msapato, Counting the number of tau-exceptional sequences over Nakayama algebras, arXiv:2002.12194 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Tree Leaf.
Index entries for sequences related to trees

Cf. A003227, A003228, A055314, A055540, A055897.

Essentially the same as A061302.

[0,2] cat [n*(n-1)^(n-2): n in [3..10]]; // G. C. Greubel, Nov 11 2017

Join[{0,2}, Table[Sum[n!/k! StirlingS2[n-2,n-k] k, {k,2,n-1}], {n,3,20}]] (* Geoffrey Critzer, Nov 22 2011 *)
Join[{0,2}, Table[n*(n-1)^(n-2), {n,3,50}]] (* or *) Rest[With[{nmax = 40}, CoefficientList[Series[-x*LambertW[-x], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Nov 11 2017 *)

for(n=1, 30, print1(if(n==1, 0, if(n==2, 2, n*(n-1)^(n-2))), ", ")) \\ G. C. Greubel, Nov 11 2017

From Vladeta Jovovic, Mar 31 2001: (Start)
a(n) = n*(n-1)^(n-2), n > 1.
E.g.f.: -x*LambertW(-x). (End)
a(n) = Sum_{k=1..n} (A055314(n, k)*k). - Christian G. Bower, Jun 12 2000
E.g.f.: x*T(x) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 07 2012

More terms from Christian G. Bower, Jun 12 2000

A327371 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 2, 0, 5, 1, 3, 1, 1, 16, 6, 7, 2, 3, 0, 78, 35, 25, 8, 7, 2, 1, 588, 260, 126, 40, 20, 6, 4, 0, 8047, 2934, 968, 263, 92, 25, 13, 3, 1, 205914, 53768, 11752, 2434, 596, 140, 47, 12, 5, 0, 10014882, 1707627, 240615, 34756, 5864, 1084, 256, 58, 21, 4, 1

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
     1;
     1,    0;
     1,    0,   1;
     2,    0,   2,   0;
     5,    1,   3,   1,  1;
    16,    6,   7,   2,  3,  0;
    78,   35,  25,   8,  7,  2,  1;
   588,  260, 126,  40, 20,  6,  4, 0;
  8047, 2934, 968, 263, 92, 25, 13, 3, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Gus Wiseman, The graphs counted in row 5 (isolated vertices not shown).

Row sums are A000088.
Row sums without the first column are A141580.
Columns k = 0..2 are A004110, A325115, A325125.
Column k = n is A059841.
Column k = n - 1 is A028242.
The labeled version is A327369.
The covering case is A327372.
Cf. A055540, A059167, A245797, A294217, A327227, A327370.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
G(n)={sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * prod(i=1, #p, (1 - x^p[i])/(1 - (x*y)^p[i]) + O(x*x^(n-k)))); x^k*s/k!)*(1-x^2*y)/(1-x^2*y^2)}
T(n)={my(v=Vec(G(n))); vector(#v, n, Vecrev(v[n], n))}
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 22 2021

Column-wise partial sums of A327372.

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2019

cp's OEIS Frontend

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

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A055541 Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.

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Magma

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A327371 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

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