A245501 -id:A245501 - 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

A034699 Largest prime power factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79

Offset: 1

David W. Wilson

n divides lcm(1, 2, ..., a(n)).
a(n) = A210208(n,A073093(n)) = largest term of n-th row in A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) = smallest m > 0 such that n divides A003418(m). - Thomas Ordowski, Nov 15 2013
a(n) = n when n is a prime power (A000961). - Michel Marcus, Dec 03 2013
Conjecture: For all n between two consecutive prime numbers, all a(n) are different. - I. V. Serov, Jun 19 2019
Disproved with between p=prime(574) = 4177 and prime(575) = 4201, a(4180) = a(4199) = 19. See A308752. - Michel Marcus, Jun 19 2019
Conjecture: For any N > 0, there exist numbers n and m, N < n < n+a(n) <= m, such that all n..m are composite and a(n) = a(m). - I. V. Serov, Jun 21 2019
Conjecture: For all n between two consecutive prime numbers, all (-1)^n*a(n) are different. Checked up to 5*10^7. - I. V. Serov, Jun 23 2019
Disproved: between p = prime(460269635) = 10120168277 and p = prime(460269636) = 10120168507 the numbers n = 10120168284 and m = 10120168498 form a pair such that (-1)^n*a(n) = (-1)^m*a(m) = 107. - L. Joris Perrenet, Jan 05 2020
a(n) = cardinality of smallest set on which idempotence of order n+1 (f^{n+1} = f) differs from idempotence of order e for 2 <= e <= n (see von Eitzen link for proof); derivable from A245501. - Mark Bowron, May 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Hagen von Eitzen, Least k to distinguish f^n = f on {1,...,k} from f^e = f for every 1 < e < n is largest prime power factor of n-1?, Math Stack Exchange.

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A245501, A284600, A308752.

Numbers n ordered by a(n): A305325.

a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

f[n_] := If[n == 1, 1, Max[ #[[1]]^#[[2]] & /@ FactorInteger@n]]; Array[f, 79] (* Robert G. Wilson v, Sep 02 2006 *)
Array[Max[Power @@@ FactorInteger@ #] &, 79] (* Michael De Vlieger, Jul 26 2018 *)

a(n) = if(1==n,n,my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ Charles R Greathouse IV, Nov 20 2012, check for a(1) added by Antti Karttunen, Aug 06 2018

A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018

from sympy import factorint
def A034699(n): return max((p**e for p, e in factorint(n).items()), default=1) # Chai Wah Wu, Apr 17 2023

If n = p_1^e_1 *...* p_k^e_k, p_1 < ... < p_k primes, then a(n) = Max_i p_i^e_i.
a(n) = A088387(n)^A088388(n). - Antti Karttunen, Jul 22 2018
a(n) = n/A284600(n) = n - A081805(n) = A034684(n) + A100574(n). - Antti Karttunen, Aug 06 2018
a(n) = a(m) iff m = d*a(n), where d is a divisor of A038610(a(n)). - I. V. Serov, Jun 19 2019

A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

Original entry on oeis.org

1, 1, 4, 19, 110, 751, 5902, 52165, 509588, 5437729, 62828306, 780287839, 10351912276, 145944541159, 2176931651546, 34225419288421, 565282627986368, 9779830102138945, 176776613812205074, 3330780287838743575

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

Number of functions f from a set of size n to itself such that f(f(f(x))) = f(x). - Joel B. Lewis, Dec 12 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A000949 - A000951, A060905 - A060913.

Column k=3 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 18 2012 *)

a(n):=sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Aug 20 2010 */

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 2.

a(n) = sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n), n>0. - Vladimir Kruchinin, Aug 20 2010

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

Original entry on oeis.org

1, 1, 3, 12, 73, 556, 4737, 44122, 453441, 5186664, 65671201, 906052654, 13418086497, 211472682604, 3535616946513, 62621439810066, 1172370604136833, 23118679430573008, 478329265510033473, 10349724555927678934, 233633352312272612001, 5492655756487132979796

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} to itself such that f(x)=f^4(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

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

Cf. A000949 - A000951, A060905 - A060913.

Column k=4 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^3/3],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 3.

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).

Original entry on oeis.org

1, 1, 4, 19, 116, 901, 8422, 89755, 1061048, 13746169, 193901066, 2965146559, 48946004956, 867463969789, 16405240966766, 329147315037811, 6973157545554128, 155446026607476145, 3636697161715448914, 89099916704329731895, 2281451214192505136516

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} into itself such that f(x) = f^5(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

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

Cf. A000949 - A000951, A060905 - A060913.

Column k=5 of A245501.

egf:= exp(x*exp(x)+x^2*exp(x)^2/2+x^4*exp(x)^4/4):
a:= n-> n!*coeff(series(egf, x, n+11), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2014

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2+a^4/4],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 4.

A245502 Number of endofunctions f on [n] such that f^6(i) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 3, 10, 41, 220, 1921, 25474, 384113, 5785192, 85295105, 1244587246, 18399553801, 286188407116, 4920039471089, 96938148715090, 2172951409994081, 53076309208887760, 1351935357372173953, 35000099470802409814, 914038174404012874745, 24209598764577808155316

Offset: 0

Alois P. Heinz, Jul 24 2014

nonn

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

Column k=6 of A245501.

egf:= exp(x*exp(x)+x^5*exp(x)^5/5):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(x*exp(x)+x^5*exp(x)^5/5).

A245503 Number of endofunctions f on [n] such that f^7(i) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 4, 21, 142, 1131, 10422, 112015, 1396516, 19702665, 306547426, 5177584809, 94168757916, 1833590759131, 38009888806450, 835064706091431, 19389119390412448, 475039405292487825, 12263445905182200882, 332913448508180564521, 9478703390668813103836

Offset: 0

Alois P. Heinz, Jul 24 2014

nonn

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

Column k=7 of A245501.

egf:= exp(x*exp(x)+x^2*exp(x)^2/2+x^3*exp(x)^3/3+x^6*exp(x)^6/6):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(x*exp(x)+x^2*exp(x)^2/2+x^3*exp(x)^3/3+x^6*exp(x)^6/6).

A245504 Number of endofunctions f on [n] such that f^8(i) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 3, 10, 41, 196, 1057, 7042, 87473, 2004328, 50881121, 1200507694, 26179091017, 538344201292, 10635924720449, 204889902343666, 3902206399428833, 74966895289570768, 1511315984060460865, 34267718094037895638, 937490346932078590841, 30931366331911427839156

Offset: 0

Alois P. Heinz, Jul 24 2014

nonn

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

Column k=8 of A245501.

egf:= exp(x*exp(x)+x^7*exp(x)^7/7):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);

With[{nn=30},CoefficientList[Series[Exp[x Exp[x]+(x^7 Exp[x]^7)/7],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2016 *)

E.g.f.: exp(x*exp(x)+x^7*exp(x)^7/7).

A245505 Number of endofunctions f on [n] such that f^9(i) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 4, 19, 116, 901, 8422, 89755, 1066088, 14154409, 212952266, 3646226959, 69912304156, 1460516349709, 32480282189006, 759186766079011, 18569220500830928, 475114325555073745, 12719547174925542034, 356098625214192355735, 10413932958067814029316

Offset: 0

Alois P. Heinz, Jul 24 2014

nonn

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

Column k=9 of A245501.

egf:= exp(add((x*exp(x))^(2^j)/(2^j), j=0..3)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=0..3} (x*exp(x))^(2^j)/(2^j)).

A245506 Number of endofunctions f on [n] such that f^10(i) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 3, 12, 73, 556, 4737, 44122, 453441, 5226984, 69703201, 1132247854, 22865062497, 541274154604, 13810143608913, 361165919643666, 9534875308290433, 254667431232941008, 6928956222876625473, 193066959857334885334, 5521621116955849623201

Offset: 0

Alois P. Heinz, Jul 24 2014

nonn

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

Column k=10 of A245501.

egf:= exp(add((x*exp(x))^(3^j)/(3^j), j=0..2)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f: exp(Sum_{j=0..2} (x*exp(x))^(3^j)/(3^j)).

cp's OEIS Frontend

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

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A034699 Largest prime power factor of n.

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A060905 Expansion of e.g.f. exp(x*exp(x) + 1/2*x^2*exp(x)^2).

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A060906 E.g.f.: exp(x*exp(x) + 1/3*x^3*exp(x)^3).

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A060907 E.g.f.: exp(x*exp(x) + 1/2*x^2*exp(x)^2 + 1/4*x^4*exp(x)^4).

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A245502 Number of endofunctions f on [n] such that f^6(i) = f(i) for all i in [n].

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Maple

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A245503 Number of endofunctions f on [n] such that f^7(i) = f(i) for all i in [n].

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A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).