A350452 -id:A350452 - cp's OEIS frontend

A000435 Normalized total height of all nodes in all rooted trees with n labeled nodes.

0, 1, 8, 78, 944, 13800, 237432, 4708144, 105822432, 2660215680, 73983185000, 2255828154624, 74841555118992, 2684366717713408, 103512489775594200, 4270718991667353600, 187728592242564421568, 8759085548690928992256, 432357188322752488126152, 22510748754252398927872000

Offset: 1

N. J. A. Sloane

This is the sequence that started it all: the first sequence in the database!
The height h(V) of a node V in a rooted tree is its distance from the root. a(n) = Sum_{all nodes V in all n^(n-1) rooted trees on n nodes} h(V)/n.
In the trees which have [0, n-1] = (0, 1, ..., n-1) as their ordered set of nodes, the number of nodes at distance i from node 0 is f(n,i) = (n-1)...(n-i)(i+1)n^(j-1), 0 <= i < n-1, i+j = n-1 (and f(n,n-1) = (n-1)!): (n-1)...(n-i) counts the words coding the paths of length i from any node to 0, n^(j-1) counts the Pruefer codes of the rest, words build by iterated deletion of the greater node of degree 1 ... except the last one, (i+1), necessary pointing at the path. If g(n,i) = (n-1)...(n-i)n^j, i+j = n-1, f(n,i) = g(n,i) - g(n,i+1), g(n,i) = Sum_{k>=i} f(n,k), the sequence is Sum_{i=1..n-1} g(n,i). - Claude Lenormand (claude.lenormand(AT)free.fr), Jan 26 2001
If one randomly selects one ball from an urn containing n different balls, with replacement, until exactly one ball has been selected twice, the probability that this ball was also the second ball to be selected once is a(n)/n^n. See also A001865. - Matthew Vandermast, Jun 15 2004
a(n) is the number of connected endofunctions with no fixed points. - Geoffrey Critzer, Dec 13 2011
a(n) is the number of weakly connected simple digraphs on n labeled nodes where every node has out-degree 1. A digraph where all out-degrees are 1 can be called a functional digraph due to the correspondence with endofunctions. - Andrew Howroyd, Feb 06 2024

			For n = 3 there are 3^2 = 9 rooted labeled trees on 3 nodes, namely (with o denoting a node, O the root node):
   o
   |
   o     o   o
   |      \ /
   O       O
The first can be labeled in 6 ways and contains nodes at heights 1 and 2 above the root, so contributes 6*(1+2) = 18 to the total; the second can be labeled in 3 ways and contains 2 nodes at height 1 above the root, so contributes 3*2=6 to the total, giving 24 in all. Dividing by 3 we get a(3) = 24/3 = 8.
For n = 4 there are 4^3 = 64 rooted labeled trees on 4 nodes, namely (with o denoting a node, O the root node):
   o
   |
   o     o        o   o
   |     |         \ /
   o     o   o      o     o o o
   |      \ /       |      \|/
   O       O        O       O
  (1)     (2)      (3)     (4)
Tree (1) can be labeled in 24 ways and contains nodes at heights 1, 2, 3 above the root, so contributes 24*(1+2+3) = 144 to the total;
tree (2) can be labeled in 24 ways and contains nodes at heights 1, 1, 2 above the root, so contributes 24*(1+1+2) = 96 to the total;
tree (3) can be labeled in 12 ways and contains nodes at heights 1, 2, 2 above the root, so contributes 12*(1+2+2) = 60 to the total;
tree (4) can be labeled in 4 ways and contains nodes at heights 1, 1, 1 above the root, so contributes 4*(1+1+1) = 12 to the total;
giving 312 in all. Dividing by 4 we get a(4) = 312/4 = 78.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 100 terms from T. D. Noe)
Vijayakumar Ambat, Article in the Malayalam newspaper Ayala Manorama - Padhippura, 12 June 2015, that mentions the OEIS, and in particular this sequence.
V. I. Arnold, Topological classification of complex trigonometric polynomials and the combinatorics of graphs with the same number of edges and vertices, Functional Anal. Appl., 30 (1996), 1-17.
Shalosh B. Ekhad and Doron Zeilberger, Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees, arXiv:1607.05776 [math.CO], 2016.
Shalosh B. Ekhad and Doron Zeilberger, Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees, Version on DZ's home page with more links; Local copy, pdf file only, no active links
I. P. Goulden and D. M. Jackson, A proof of a conjecture for the number of ramified coverings of the sphere by the torus, arXiv:math/9902009 [math.AG], 1999.
I. P. Goulden, D. M. Jackson, and A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera, arXiv:math/9902125 [math.AG], 1999.
I. P. Goulden, D. M. Jackson, and A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera Ann. Comb. 4 (2000), no. 1, 27-46. (See Theorem 1.1.)
Brady Haran, The Number Collector (with Neil Sloane), Numberphile Podcast (2019)
Andrew Lohr and Doron Zeilberger, On the limiting distributions of the total height on families of trees, Integers (2018) 18, Article #A32.
T. Kyle Petersen, Exponential generating functions and Bell numbers, Inquiry-Based Enumerative Combinatorics (2019) Chapter 7, Undergraduate Texts in Mathematics, Springer, Cham, 98-99.
A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7).
Marko Riedel et al., Connected endofunctions with no fixed points, Mathematics Stack Exchange, Dec 2014.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970
J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
N. J. A. Sloane, Page from 1964 notebook showing start of OEIS [includes A000027, A000217, A000292, A000332, A000389, A000579, A000110, A007318, A000058, A000215, A000289, A000324, A234953 (= A001854(n)/n), A000435, A000169, A000142, A000272, A000312, A000111]
N. J. A. Sloane, Cover of same notebook
N. J. A. Sloane, Lengths of Cycle Times in Random Neural Networks, Ph. D. Dissertation, Cornell University, February 1967; also Report No. 10, Cognitive Systems Research Program, Cornell University, 1967. This sequence appears on page 119.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Illustration of a(3) and a(4)
Yukun Yao and Doron Zeilberger, An Experimental Mathematics Approach to the Area Statistic of Parking Functions, arXiv:1806.02680 [math.CO], 2018.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 3.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A001863, A001864, A001854, A002862 (unlabeled version), A234953, A259334.

Column k=1 of A350452.

A000435 := n-> (n-1)!*add (n^k/k!, k=0..n-2);
seq(simplify((n-1)*GAMMA(n-1,n)*exp(n)),n=1..20); # Vladeta Jovovic, Jul 21 2005

f[n_] := (n - 1)! Sum [n^k/k!, {k, 0, n - 2}]; Array[f, 18] (* Robert G. Wilson v, Aug 10 2010 *)
nx = 18; Rest[ Range[0, nx]! CoefficientList[ Series[ LambertW[-x] - Log[1 + LambertW[-x]], {x, 0, nx}], x]] (* Robert G. Wilson v, Apr 13 2013 *)

x='x+O('x^30); concat(0, Vec(serlaplace(lambertw(-x)-log(1+lambertw(-x))))) \\ Altug Alkan, Sep 05 2018

A000435(n)=(n-1)*A001863(n) \\ M. F. Hasler, Dec 10 2018

from math import comb
def A000435(n): return ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n # Chai Wah Wu, Apr 25-26 2023

a(n) = (n-1)! * Sum_{k=0..n-2} n^k/k!.
a(n) = A001864(n)/n.
E.g.f.: LambertW(-x) - log(1+LambertW(-x)). - Vladeta Jovovic, Apr 10 2001
a(n) = A001865(n) - n^(n-1).
a(n) = A001865(n) - A000169(n). - Geoffrey Critzer, Dec 13 2011
a(n) ~ sqrt(Pi/2)*n^(n-1/2). - Vaclav Kotesovec, Aug 07 2013
a(n)/A001854(n) ~ 1/2 [See Renyi-Szekeres, (4.7)]. Also a(n) = Sum_{k=0..n-1} k*A259334(n,k). - David desJardins, Jan 20 2017
a(n) = (n-1)*A001863(n). - M. F. Hasler, Dec 10 2018

Additional references from Valery A. Liskovets
Editorial changes by N. J. A. Sloane, Feb 03 2012
Edited by M. F. Hasler, Dec 10 2018

A065440 a(n) = (n-1)^n.

Original entry on oeis.org

1, 0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

Len Smiley, Nov 17 2001

a(n) is the number of functions from {1,2,...,n} into {1,2,...,n} that have no fixed points.

The probability that a random function from {1,2,...,n} into {1,2,...,n} has no fixed point is equal to a(n)/n^n; it tends to 1/e when n tends to infinity. - Robert FERREOL, Mar 29 2017

Harry J. Smith, Table of n, a(n) for n = 0..100
Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.

Essentially the same as A007778 - note T(x) = -W(-x).
Column k=0 of A055134.
Row sums of A350452.
Cf. A158497, A284458.

A065440:= func< n | (n-1)^n >;
[A065440(n): n in [0..30]]; // G. C. Greubel, Mar 23 2025

Table[(n-1)^n,{n,0,20}] (* Harvey P. Dale, Jan 03 2015 *)

a(n) = { (n - 1)^n } \\ Harry J. Smith, Oct 19 2009

def A065440(n): return (n-1)**n
print([A065440(n) for n in range(31)]) # G. C. Greubel, Mar 23 2025

a(n) = A007778(n-1).
E.g.f.: x/(T(x)*(1-T(x))) (where T(x) is Euler's tree function, the E.g.f. for n^(n-1)) (see A000169).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*n^(n-k). - Robert FERREOL, Mar 28 2017
a(n) = Sum_{k=0..n} (-1)^k*binomial(n+2,k+2)*(k+1)*(2*k+n+3)^n. - Vladimir Kruchinin, Aug 13 2025

A350446 Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 3, 1, 16, 11, 125, 128, 3, 1296, 1734, 95, 16807, 27409, 2425, 15, 262144, 499400, 61054, 945, 4782969, 10346328, 1605534, 42280, 105, 100000000, 240722160, 44981292, 1706012, 11025, 2357947691, 6222652233, 1351343346, 67291910, 763875, 945

Offset: 0

Alois P. Heinz, Dec 31 2021

			Triangle T(n,k) begins:
           1;
           1;
           3,          1;
          16,         11;
         125,        128,          3;
        1296,       1734,         95;
       16807,      27409,       2425,       15;
      262144,     499400,      61054,      945;
     4782969,   10346328,    1605534,    42280,    105;
   100000000,  240722160,   44981292,  1706012,  11025;
  2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives A000272(n+1).
Row sums give A000312.
T(2n,n) gives A001147.
Cf. A055134, A060281, A349454, A350452.
Cf. A136394, A190314, A217701.

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(i)), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]];
T[n_] :=  With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

From Mélika Tebni, Mar 23 2023: (Start)
E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.
T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).
Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.
Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)

cp's OEIS Frontend

A000435 Normalized total height of all nodes in all rooted trees with n labeled nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A065440 a(n) = (n-1)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A350446 Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula