A000435 -id:A000435 - cp's OEIS frontend

A001865 Number of connected functions on n labeled nodes.

1, 3, 17, 142, 1569, 21576, 355081, 6805296, 148869153, 3660215680, 99920609601, 2998836525312, 98139640241473, 3478081490967552, 132705415800984825, 5423640496274200576, 236389784118231290049, 10944997108429625524224, 536484538620663729658993

Offset: 1

N. J. A. Sloane

If one randomly selects a ball from an urn containing n different balls, with replacement, until exactly one ball has been selected twice, the probability that that ball was also the first ball selected once is a(n)/n^n. See also A000435. - Matthew Vandermast, Jun 15 2004

a(n) equals the permanent of the (n-1) X (n-1) matrix with n+1's along the main diagonal and 1's everywhere else. - John M. Campbell, Apr 20 2012

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 112.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
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).

Alois P. Heinz, Table of n, a(n) for n = 1..380 (first 50 terms from T. D. Noe)
H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910 [math.CO], 2013.
Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See pp. 8, 19.
Camille Combe, A geometric and combinatorial exploration of Hochschild lattices, arXiv:2007.00048 [math.CO], 2020. See p. 22.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 37
Leo Katz, Probability of indecomposability of a random mapping function, Ann. Math. Statist. 26, (1955), 512-517.
John Riordan, Letter to N. J. A. Sloane, Aug. 1970
Frank Schmidt and Rodica Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34.
Bernd Sturmfels and Ngoc Mai Tran, Combinatorial Types of Tropical Eigenvectors, arXiv:1105.5504 [math.CO], 2011-2012.

a(n) = A000435(n) + n^(n-1). See also A063169.

Column k=1 of A060281.

spec := [B, {A=Prod(Z,Set(A)), B=Cycle(A)}, labeled]; [seq(combstruct[count](spec,size=n), n=0..20)];
seq(simplify(GAMMA(n,n)*exp(n)),n=1..20); # Vladeta Jovovic, Jul 21 2005

t=Sum[n^(n-1)x^n/n!,{n,1,20}];
Range[0,20]! CoefficientList[Series[Log[1/(1-t)]+1,{x,0,20}],x] (* Geoffrey Critzer, Mar 12 2011 *)
f[n_] := Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]; Array[f, 18] (* Robert G. Wilson v *)
a[n_] := Exp[n]*Gamma[n, n]; Table[a[n] // FunctionExpand, {n, 1, 18}] (* Jean-François Alcover, May 13 2013, after Vladeta Jovovic *)

a(n)=if(n<0,0,n!*sum(k=1,n,n^(n-k-1)/(n-k)!))

a(n)=(1/n)*sum(k=1,n,binomial(n,k)*(n-k)^(n-k)*k^k) \\ Paul D. Hanna, Jul 04 2013

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, (k*x)^k/k!)))) \\ Seiichi Manyama, May 27 2019

from math import comb
def A001865(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 + n**(n-1) # Chai Wah Wu, Apr 25-26 2023

a(n) = Sum_{k=1..n} n!*n^(n-k-1) / (n-k)!.
E.g.f.: -log(1+LambertW(-x)). - Vladeta Jovovic, Apr 11 2001
E.g.f. satisfies 0=2y'^4+2y''^2-y'''y'-y''y'^2. - Michael Somos, Aug 23 2003
Integral representation in terms of the incomplete Gamma function: a(n) = exp(n+1)*Gamma(n+1,n+1) = exp(n+1)*Integral_{x=n+1..oo} x^n exp(-x) dx.
Asymptotics: sqrt(Pi*n/2)*n^(n-1). - N-E. Fahssi, Jan 25 2008, corrected by Vaclav Kotesovec, Nov 27 2012
a(n) = exp(1)*Integral_{x=1..oo} (n+x)^n*exp(-x) dx. - Gerald McGarvey, Apr 16 2008
a(n) = (1/n) * Sum_{k=1..n} C(n,k) * (n-k)^(n-k) * k^k. - Paul D. Hanna, Jul 04 2013
From Peter Bala, Jun 29 2016: (Start)
It appears that a(n) = (n-1)!*( e^n - Sum_{k >= 0} n^(n + k)/(n + k)! ) = (n-1)!*( e^n - Sum_{k >= 0} k^2*n^(n + k - 1)/(n + k)! ).
Note that (n-1)!*( e^n - Sum_{k >= 0} k^3*n^(n + k - 1)/(n + k)! ) also appears to be an integer sequence beginning [1, 5, 37, 370, 4681, 71736, 1292005, ...]. (End)
a(n) = Sum_{k=1..n} (n!/(n-k)!) * k^2 * n^(n-k-2). - Brian P Hawkins, Feb 07 2024

More terms from James Sellers, May 23 2000

A001864 Total height of rooted trees with n labeled nodes.

Original entry on oeis.org

0, 2, 24, 312, 4720, 82800, 1662024, 37665152, 952401888, 26602156800, 813815035000, 27069937855488, 972940216546896, 37581134047987712, 1552687346633913000, 68331503866677657600, 3191386068123595166656, 157663539876436721860608

Offset: 1

N. J. A. Sloane

a(n) is the total number of nonrecurrent elements mapped into a recurrent element in all functions f:{1,2,...,n}->{1,2,...,n}. a(n) = Sum_{k=1..n-1} A216971(n,k)*k. - Geoffrey Critzer, Jan 01 2013

a(n) is the sum of the lengths of all cycles over all functions f:{1,2,...,n}->{1,2,...,n}. Fixed points are taken to have length zero. a(n) = Sum_{k=2..n} A066324(n,k)*(k-1). - Geoffrey Critzer, Aug 19 2013

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).

T. D. Noe, Table of n, a(n) for n = 1..100
Hien D. Nguyen and G. J. McLachlan, Progress on a Conjecture Regarding the Triangular Distribution, arXiv preprint arXiv:1607.04807 [stat.OT], 2016.
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, Illustration of terms a(3) and a(4) in A000435
D. Zvonkine, Home Page
D. Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere, arXiv:0403092v2 [math.AG], 2004.
D. Zvonkine, Enumeration of ramified coverings of the sphere and 2-dimensional gravity, arXiv:math/0506248 [math.AG], 2005.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, vol. 7 (2007), no. 1, 135-162.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000435, A001863.

A001864 := proc(n) local k; add(n!*n^k/k!, k=0..n-2); end;

Table[Sum[Binomial[n,k](n-k)^(n-k) k^k,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Oct 10 2011 *)
a[n_] := n*(n-1)*Exp[n]*Gamma[n-1, n] // Round; Table[a[n], {n, 1, 18}]  (* Jean-François Alcover, Jun 24 2013 *)

a(n)=sum(k=1,n-1,binomial(n,k)*(n-k)^(n-k)*k^k)

from math import comb
def A001864(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) # Chai Wah Wu, Apr 25-26 2023

a(n) = n*A000435(n).
E.g.f: (LambertW(-x)/(1+LambertW(-x)))^2. - Vladeta Jovovic, Apr 10 2001
a(n) = Sum_{k=1..n-1} binomial(n, k)*(n-k)^(n-k)*k^k. - Benoit Cloitre, Mar 22 2003
a(n) ~ sqrt(Pi/2)*n^(n+1/2). - Vaclav Kotesovec, Aug 07 2013
a(n) = n! * Sum_{k=0..n-2} n^k/k!. - Jianing Song, Aug 08 2022

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A001863 Normalized total height of rooted trees with n nodes.

Original entry on oeis.org

0, 1, 4, 26, 236, 2760, 39572, 672592, 13227804, 295579520, 7398318500, 205075286784, 6236796259916, 206489747516416, 7393749269685300, 284714599444490240, 11733037015160276348, 515240326393584058368, 24019843795708471562564, 1184776250223810469888000

Offset: 1

N. J. A. Sloane

a(n) is the number of partial functions f from [n-1] into [n-1] such that f^k(1) is undefined for some k>=1. - Geoffrey Critzer, Mar 05 2022

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).

G. C. Greubel, Table of n, a(n) for n = 1..385 (terms 1..100 from T. D. Noe)
H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910 [math.CO], 2013.
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.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000435, A000169, A001864.

[0] cat [&+[Factorial(n-2)*n^k div Factorial(k): k in [0..n-2]]: n in [2..24]]; // Vincenzo Librandi, Dec 10 2018

A001863 := n->add((n-2)!*n^k/k!, k=0..n-2); # for n>1. Equals A001864(n)/(n^2-n)
seq(simplify(GAMMA(n-1,n)*exp(n)),n=2..20); # Vladeta Jovovic, Jul 21 2005

a[n_] := Sum[(n-2)!*n^k/k!, {k, 0, n-2}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Oct 09 2012, from Maple *)
Table[Sum[(n-2)! n^k/k!,{k,0,n-2}],{n,30}] (* Harvey P. Dale, Jun 19 2016 *)

apply( A001863(n)=sum(k=0,n-2,(n-2)!/k!*n^k), [1..20]) \\ This defines the function A001863; apply(...) provides a check and illustration. - G. C. Greubel, Nov 14 2017, edited by M. F. Hasler, Dec 09 2018

from math import comb
def A001863(n): return 0 if n<2 else ((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//(n-1) # Chai Wah Wu, Apr 25-26 2023

E.g.f.: -exp(1)*x*(Ei(-1-LambertW(-x))-Ei(-1)) - LambertW(-x) + log(1+LambertW(-x)). - Vladeta Jovovic, Sep 29 2003
a(n)*(n-1) = A000435(n). - M. F. Hasler, Dec 10 2018
E.g.f.: x*diff(A000169(x),x)^2. - Vladimir Kruchinin, Jun 07 2020
a(n) = (n-2)! * Sum_{k=0..n-2} n^k/k! for n > 1. - Jianing Song, Aug 08 2022

A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

Original entry on oeis.org

2, 3, 6, 4, 36, 24, 5, 200, 300, 120, 6, 1170, 3360, 2520, 720, 7, 7392, 38850, 43680, 22680, 5040, 8, 50568, 475776, 757680, 551040, 221760, 40320, 9, 372528, 6231960, 13747104, 12836880, 7136640, 2358720, 362880, 10, 2936070, 87530400, 264181680

Offset: 2

N. J. A. Sloane

			2;
3,    6;
4,   36,    24;
5,  200,   300,   120;
6, 1170,  3360,  2520,   720;
7, 7392, 38850, 43680, 22680, 5040;

Alois P. Heinz, Rows n = 2..101, flattened
Marko Riedel, Counting the number of rooted trees of a certain height
Marko Riedel, Maple code for sequence (EGF)
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. [broken link]
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A001854, A234953, A000435, A236396.

gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:
A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:
T:= (n, d)-> A(n, d) -A(n, d-1):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Sep 21 2012

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k - 1]]; a[n_, k_] := n!*Coefficient[ Series[gf[k], {x, 0, n + 1}], x, n]; t[n_, d_] := a[n, d] - a[n, d - 1]; Table[t[n, d], {n, 2, 12}, {d, 1, n - 1}] // Flatten (* Jean-François Alcover, Jan 15 2013, translated from Alois P. Heinz's Maple program *)

Riordan reference gives recurrence.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

A001854 Total height of all rooted trees on n labeled nodes.

Original entry on oeis.org

0, 2, 15, 148, 1785, 26106, 449701, 8927192, 200847681, 5053782070, 140679853941, 4293235236324, 142553671807729, 5116962926162738, 197459475792232725, 8152354312656732976, 358585728464893234305, 16741214317684425260142, 826842457727306803110997, 43073414675338753123113980

Offset: 1

N. J. A. Sloane

nonn

Take any one of the n^(n-1) rooted trees on n labeled nodes, compute its height (maximal edge distance to root), sum over all trees.

Theorem [Renyi-Szekeres, (4,7)]. The average height if the tree is chosen at random is sqrt(2*n*Pi). - David desJardins, Jan 20 2017

Rényi, A., and G. Szekeres. "On the height of trees." Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7).
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).

Alois P. Heinz, Table of n, a(n) for n = 1..387
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000435, A034855, A236396.

Also A234953(n) = a(n)/n.

nn=20;a=NestList[ x Exp[#]&,x,nn];f[list_]:=Sum[list[[i]]*i,{i,1,Length[list]}];Drop[Map[f,Transpose[Table[Range[0,nn]!CoefficientList[Series[a[[i+1]]-a[[i]],{x,0,nn}],x],{i,1,nn-1}]]],1]  (* Geoffrey Critzer, Mar 14 2013 *)

a(n) = Sum_{k=1..n-1} A034855(n,k)*k. - Geoffrey Critzer, Mar 14 2013

A000435(n)/a(n) ~ 1/2 (see A000435 and the Renyi-Szekeres result mentioned in the Comments). - David desJardins, Jan 20 2017

More terms from Geoffrey Critzer, Mar 14 2013

A133297 a(n) = n!Sum_{k=1..n} (-1)^(k+1)n^(n-k-1)/(n-k)!.

Original entry on oeis.org

0, 1, 1, 5, 34, 329, 4056, 60997, 1082320, 22137201, 512801920, 13269953861, 379400765184, 11877265764025, 404067857880064, 14843708906336325, 585606019079612416, 24693567694861202273, 1108343071153648926720, 52757597474618636748421, 2654611611461360017408000

Offset: 0

Vladeta Jovovic, Oct 17 2007

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A001865 (Gamma(n, n)/exp(-n)).

Cf. A000272, A000435, A277458, A292977.

a:= function(n)
    if n=0 then return 0;
    else return Factorial(n)*Sum([1..n], k-> (-1)^(k+1)*n^(n-k-1)/Factorial(n-k));
    fi;
  end;
List([0..25], n-> a(n) ); # G. C. Greubel, Aug 02 2019

a:= func< n | n eq 0 select 0 else Factorial(n)*(&+[(-1)^(k+1)*n^(n-k-1)/Factorial(n-k): k in [1..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Aug 02 2019

Table[n!*Sum[(-1)^(k+1)*n^(n-k-1)/(n-k)!, {k,n}], {n,0,25}] (* Stefan Steinerberger, Oct 19 2007 *)
With[{m=25}, CoefficientList[Series[Log[1-LambertW[-x]], {x,0,m}], x]*Range[0,m]!] (* G. C. Greubel, Aug 02 2019 *)

my(x='x+O('x^25)); concat([0], Vec(serlaplace( log(1-lambertw(-x)) ))) \\ G. C. Greubel, Aug 02 2019

def a(n):
    if (n==0): return 0
    else: return factorial(n)*sum((-1)^(k+1)*n^(n-k-1)/factorial(n-k) for k in (1..n))
[a(n) for n in (0..25)] # G. C. Greubel, Aug 02 2019

E.g.f.: log(1-LambertW(-x)).
a(n) ~ n^(n-1)/2. - Vaclav Kotesovec, Sep 25 2013
Conjecture: a(n) = (n-1)!*( Sum_{k >= 0} (-1)^k * n^(n+k)/(n+k)! - (-1/e)^n ) for n >= 1. Cf. A000435. - Peter Bala, Jul 23 2021
From Thomas Scheuerle, Nov 17 2023: (Start)
This conjecture is true. Let "gamma" be the lower incomplete gamma function: gamma(n, x) = (n-1)! (1 - exp(-x)*Sum_{k = 0..n-1} x^k/k! ), then we can get the upper incomplete gamma function Gamma(n, x) = gamma(n, oo) - gamma(n, x). By inserting according the formula below, we will obtain the formula from Peter Bala.
a(n) = (-1)^(n+1)*Gamma(n, -n)/exp(n) = (-1)^(n+1)*A292977(n-1, n), for n > 0, where Gamma is the upper incomplete gamma function. (End)

More terms from Stefan Steinerberger, Oct 19 2007

A234953 Normalized total height of all rooted trees on n labeled nodes.

Original entry on oeis.org

0, 1, 5, 37, 357, 4351, 64243, 1115899, 22316409, 505378207, 12789077631, 357769603027, 10965667062133, 365497351868767, 13163965052815515, 509522144541045811, 21093278144993719665, 930067462093579181119, 43518024090910884374263, 2153670733766937656155699

Offset: 1

N. J. A. Sloane, Jan 14 2014

nonn

Equals A001854(n)/n. That is, similar to A001854, except here the root always has the fixed label 1.

This was in one of my thesis notebooks from 1964 (see the scans in A000435), but because it wasn't of central importance it was never added to the OEIS.

Alois P. Heinz, Table of n, a(n) for n = 1..387

Cf. A001854, A034855, A235595, A236396.

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; a[n_] := Sum[k*(a[n, k] - a[n, k-1]), {k, 1, n-1}]/n; Array[a, 20] (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
def a(n): return sum([k*T(n, k) for k in range(1, n)])
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 26 2017

a(n) = Sum_{k=1..n-1} k*A034855(n,k)/n = Sum_{k=1..n-1} k*A235595(n,k).

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 18, 6, 0, 0, 192, 72, 48, 0, 0, 2500, 960, 720, 540, 0, 0, 38880, 15000, 11520, 9720, 7680, 0, 0, 705894, 272160, 210000, 181440, 161280, 131250, 0, 0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0

Offset: 0

Peter Luschny, Jan 11 2024

A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.

			Triangle starts:
  [0] [0]
  [1] [0,        0]
  [2] [0,        2,       0]
  [3] [0,       18,       6,       0]
  [4] [0,      192,      72,      48,      0]
  [5] [0,     2500,     960,     720,     540,       0]
  [6] [0,    38880,   15000,   11520,    9720,    7680,       0]
  [7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
  [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]

John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.

T(n, 1) = A066274(n) for n >= 1.
T(n, 1)/(n - 1) = A000169(n) for n >= 2.
T(n, n - 1) = 2*A081133(n) for n >= 1.
Sum_{k=0..n} T(n, k) = A001864(n).
(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).

A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)

def T(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])

A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

Original entry on oeis.org

1, 1, 2, 1, 9, 6, 1, 40, 60, 24, 1, 195, 560, 420, 120, 1, 1056, 5550, 6240, 3240, 720, 1, 6321, 59472, 94710, 68880, 27720, 5040, 1, 41392, 692440, 1527456, 1426320, 792960, 262080, 40320, 1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880, 1, 2237920, 119723130, 490458240, 691331760, 570810240, 323114400, 125798400, 30844800, 3628800

Offset: 2

N. J. A. Sloane, Jan 14 2014

T(n,k) is the number of forests of labeled rooted trees with n nodes and height k Cf. A210725. Equivalently, T(n,k) is the number of nilpotent partial functions on [n] with index k+1. - Geoffrey Critzer, Nov 26 2021

			Triangle begins:
1.
1, 2,
1, 9, 6,
1, 40, 60, 24,
1, 195, 560, 420, 120,
1, 1056, 5550, 6240, 3240, 720,
1, 6321, 59472, 94710, 68880, 27720, 5040,
1, 41392, 692440, 1527456,1426320, 792960, 262080, 40320,
1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880,
...

Alois P. Heinz, Rows n = 2..142, flattened

Cf. A000435, A034855, A001854, A210725, A234953, A235596, A236396.

Row sums give A000272.

b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*j*b(j-1, h-1)*b(n-j, h), j=1..n))
    end:
T:= (n,k)-> b(n-1, k-1)-b(n-1, k-2):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Aug 21 2017

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; t[n_, k_] := (a[n, k] - a[n, k-1])/n; Table[t[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
for n in range(2, 13): print([T(n, d) for d in  range(1, n)]) # Indranil Ghosh, Aug 26 2017, after Maple code

A234953(n) = Sum_{k=1..n} k*T(n,k).

cp's OEIS Frontend

A001865 Number of connected functions on n labeled nodes.

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A001864 Total height of rooted trees with n labeled nodes.

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A274804 The exponential transform of sigma(n).

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A001863 Normalized total height of rooted trees with n nodes.

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A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

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A001854 Total height of all rooted trees on n labeled nodes.

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A133297 a(n) = n!Sum_{k=1..n} (-1)^(k+1)n^(n-k-1)/(n-k)!.

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).