A210725 -id:A210725 - cp's OEIS frontend

A000248 Expansion of e.g.f. exp(x*exp(x)).

1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921, 18210094, 157329097, 1436630092, 13810863809, 139305550066, 1469959371233, 16184586405328, 185504221191745, 2208841954063318, 27272621155678841, 348586218389733556, 4605223387997411873

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of forests with n nodes and height at most 1.
Equivalently, number of idempotent mappings f from a set of n elements into itself (i.e., satisfying f o f = f). - Robert FERREOL, Oct 11 2007
In other words, a(n) = number of idempotents in the full semigroup of maps from [1..n] to itself. [Tainiter]
a(n) is the number of ways to select a set partition of {1,2,...,n} and then designate one element in each block (cell) of the partition.
Let set B have cardinality n. Then a(n) is the number of functions f:D->C over all partitions {D,C} of B. See the example in the Example Section below. We note that f:empty set->B is designated as the null function, whereas f:B->empty set is undefined unless B itself is empty. - Dennis P. Walsh, Dec 05 2013
In physics, a(n) would be interpreted as the number of projection operators P on S_n, i.e., ones satisfying P^2 = P. Example: a particle with a half-integer spin s has a spin space with 2s+1 base states which admits a(2s+1) linear projection operators (including the identity). These are important because they satisfy the operator identity exp(zU) = 1+(exp(z)-1)*U, valid for any complex z. - Stanislav Sykora, Nov 03 2016

			a(3)=10 since, for B={1,2,3}, we have 10 functions: 1 function of the type f:empty set->B; 6 functions of the type f:{x}->B\{x}; and 3 functions of the type f:{x,y}->B\{x,y}. - _Dennis P. Walsh_, Dec 05 2013

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

Alois P. Heinz, Table of n, a(n) for n = 0..541 (first 101 terms from T. D. Noe)
Rudolf Berghammer, Jules Desharnais, and Michael Winter, Counting Specific Classes of Relations Regarding Fixed Points and Reflexive Points, arXiv:2505.00140 [cs.DM], 2025. See p. 24.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 131.
Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.13.
Bernhard Harris and Lowell Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135.
Bernhard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois Journal of Mathematics, Volume 12, Issue 2 (1968), 264-277.
Gottfried Helms, Pascalmatrix tetrated
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117
Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms)
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974
Emre Sen, Exceptional Sequences and Idempotent Functions, arXiv:1909.05887 [math.RT], 2019.
Melvin Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373.
Haoliang Wang and Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.

First row of array A098697.
Row sums of A133399.
Column k=1 of A210725, A279636.
Column k=2 of A245501.
Cf. A005727, A048993.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Feb 01 2020

A000248 := proc(n) local k; add(k^(n-k)*binomial(n, k), k=0..n); end; # Robert FERREOL, Oct 11 2007
a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j) *(j+1) *a(n-1-j), j=0..n-1) fi end: seq(a(n), n=0..20); # Zerinvary Lajos, Mar 28 2009

CoefficientList[Series[Exp[x Exp[x]],{x,0,20}],x]*Table[n!,{n,0,20}]
a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[(Binomial[n - 1, j] + (n - 1) Binomial[n - 2, j]) a[j], {j, 0, n - 2}]; Table[a[n], {n, 0, 20}] (* David Callan, Oct 04 2013 *)
Flatten[{1,Table[Sum[Binomial[n,k]*(n-k)^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 13 2014 *)
Table[Sum[BellY[n, k, Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k); \\ Paul D. Hanna, Jun 26 2009

x='x+O('x^66); Vec(serlaplace(exp(x*exp(x)))) \\ Joerg Arndt, Oct 06 2013

# uses[bell_matrix from A264428]
B = bell_matrix(lambda k: k+1, 20)
print([sum(B.row(n)) for n in range(20)]) # Peter Luschny, Sep 03 2019

G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003
a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. - Paul D. Hanna, Jun 26 2009
G.f.: G(0) where G(k) = 1 - x*(-1+2*k*x)^(2*k+1)/((x-1+2*k*x)^(2*k+2) - x*(x-1+2*k*x)^(4*k+4)/(x*(x-1+2*k*x)^(2*k+2) - (2*x-1+2*k*x)^(2*k+3)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + exp(x)/(k+1)/(1-x/(x+(1)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} binomial(n-1,k-1)*k*a(n-k). - James East, Mar 30 2014
Asymptotics (Harris and Schoenfeld, 1968): a(n) ~ sqrt((r+1)/(2*Pi*(n+1)*(r^2+3*r+1))) * n! * exp((n+1)/(r+1)) / r^n, where r is the root of the equation r*(r+1)*exp(r) = n+1. - Vaclav Kotesovec, Jul 13 2014
a(n) = Sum_{k=0..n} A005727(k)*Stirling2(n, k). - Mélika Tebni, Jun 12 2022
More precise asymptotics: a(n) ~ n^(n + 1/2) / (sqrt(1 + 3*r + r^2) * exp(n - r*exp(r) + r/2) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2). - Vaclav Kotesovec, Feb 20 2023

In view of the multiple appearances of this sequence, I replaced the definition with the simple exponential generating function. - N. J. A. Sloane, Apr 16 2018

A000949 Number of forests with n nodes and height at most 2.

Original entry on oeis.org

1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021

Offset: 0

N. J. A. Sloane

nonn

Equivalently, the number of mappings from a set of n elements into itself where f(f(x)) = f(f(f(x))). - Chad Brewbaker, Mar 26 2014

			G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - _Michael Somos_, Jul 03 2018

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 = 0..100
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000950, A000951, A052512-A052514.

Column k=2 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *)

a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!),k,1,n-m)/m!,m,1,n-1)+1; /* Vladimir Kruchinin, May 28 2011 */

x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */

E.g.f.: exp(x*exp(x*exp(x))).

a(n) = n!*sum(m=1..n-1, sum(k=1..n-m, (k^(n-m-k)*m^k)/(k!*(n-m-k)!))/m!)+1. - Vladimir Kruchinin, May 28 2011

More terms from Vladeta Jovovic, Apr 07 2001

A000951 Number of forests with n nodes and height at most 4.

Original entry on oeis.org

1, 3, 16, 125, 1296, 16087, 229384, 3687609, 66025360, 1303751051, 28151798544, 659841763957, 16681231615816, 452357366282655, 13095632549137576, 403040561722348913, 13138626717852194976, 452179922268565180819, 16381932383826669204640

Offset: 1

N. J. A. Sloane

nonn

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
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949, A000950, A052512-A052514.

Column k=4 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x*Exp[x]]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x*exp(x))))).

More terms from Vladeta Jovovic, Apr 07 2001

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).

A000950 Number of forests with n nodes and height at most 3.

Original entry on oeis.org

1, 3, 16, 125, 1176, 12847, 160504, 2261289, 35464816, 612419291, 11539360944, 235469524237, 5170808565976, 121535533284999, 3043254281853496, 80852247370051793, 2270951670959226336, 67221368736302224819, 2091039845329887687136

Offset: 1

N. J. A. Sloane

nonn

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
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949-A000951, A052512-A052514.

Column k=3 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x)))).

More terms from Vladeta Jovovic, Apr 07 2001

A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

Original entry on oeis.org

1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10

Offset: 2

Stefano Spezia, Oct 20 2020

T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).

Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.

Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
John Riordan, Forests of labeled trees, Journal of Combinatorial Theory 5 (1968), 93-103.
Lajos Takács, On Cayley’s Formula for Counting Forests, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.
Johan Wästlund, Padlock Solitaire: A martingale trick for combinatorial enumeration, arXiv:2008.13017 [math.CO], 2020. See Section 3.

Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.

Table[k*n^(n-k-1),{n,2,11},{k,1,n-1}]//Flatten

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A000248 Expansion of e.g.f. exp(x*exp(x)).

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A000949 Number of forests with n nodes and height at most 2.

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A000951 Number of forests with n nodes and height at most 4.

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

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A000950 Number of forests with n nodes and height at most 3.

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A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

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