A279636 -id:A279636 - 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

A033462 Exponential (or "EXP") transform of squares A000290.

Original entry on oeis.org

1, 1, 5, 22, 125, 836, 6277, 52396, 479593, 4757680, 50738921, 577894604, 6990138685, 89376020800, 1203182518189, 16995248375116, 251135780602193, 3871961504546624, 62141329025501905, 1035979079450355532, 17907209511611407141, 320387246623657457056, 5924125441456047522005

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of ways to select an ordered pair from each equivalence class in each equivalence relation on {1,2,...,n}. - Geoffrey Critzer, Oct 03 2011

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=2 of A279636.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 30 2016

Range[0,20]! CoefficientList[Series[Exp[Exp[x](x+x^2)],{x,0,20}],x]
Table[Sum[BellY[n, k, Range[n]^2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

N=33;  x='x+O('x^N);
egf=exp(x*(1+x)*exp(x));
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

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

A279358 Exponential transform of the cubes A000578.

Original entry on oeis.org

1, 1, 9, 52, 413, 3916, 41077, 481384, 6198425, 86430160, 1296040841, 20763245944, 353272341061, 6353672109760, 120315348389069, 2390488408994536, 49682962883210033, 1077292416660660736, 24313317132393295633, 569937590287796925784, 13850459183086300341341

Offset: 0

Ilya Gutkovskiy, Dec 10 2016

nonn

			E.g.f.: A(x) = 1 + x/1! + 9*x^2/2! + 52*x^3/3! + 413*x^4/4! + 3916*x^5/5! + 41077*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..479
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 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 Weisstein's World of Mathematics, Exponential Transform
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A033462.

Column k=3 of A279636.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^3*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 11 2016

Range[0, 20]! CoefficientList[Series[Exp[Exp[x] (x + 3 x^2 + x^3)], {x, 0, 20}], x]

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

A279644 a(n) is the n-th term of the exponential transform of the n-th powers.

Original entry on oeis.org

1, 1, 5, 52, 1445, 104116, 16379797, 6067246144, 5270005429705, 9832425683734288, 40944833826904310921, 384044953998005246634304, 7656468877618298485395299533, 332312769563869315072667521436992, 31885558406529826494376921134421228189

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Number of labeled magmas with n elements satisfying the equation x(yz) = xz. - Andrew Howroyd, Apr 24 2023

Alois P. Heinz, Table of n, a(n) for n = 0..76
Wikipedia, Kronecker delta

Main diagonal of A279636.

Cf. A362384.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^k*b(n-j, k), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Binomial[n-1, j-1]*j^k*b[n-j, k], {j, 1, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = n! * [x^n] exp(exp(x)*(Sum_{j=0..n} Stirling2(n,j)*x^j) - delta_{0,n}).

A279637 Exponential transform of the fourth powers A000583.

Original entry on oeis.org

1, 1, 17, 130, 1445, 19676, 288517, 4768240, 86825545, 1707427792, 36133006121, 817372392464, 19631012216653, 498360729728512, 13320962518548973, 373554936371438896, 10956734043885307793, 335251566923262901760, 10675684185273726205393, 353052079426340899698736

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..452

Column k=4 of A279636.

Cf. A000583.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^4*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^4+6*x^3+7*x^2+x)).

A279638 Exponential transform of the fifth powers A000584.

Original entry on oeis.org

1, 1, 33, 340, 5261, 104116, 2133397, 49873552, 1290339353, 35858779408, 1073946466601, 34411135594144, 1169673799665637, 42024908270673472, 1589960095129885949, 63097191029229655456, 2618689624916494795313, 113366790415884862467328, 5107030221925521874906705

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..428

Column k=5 of A279636.

Cf. A000584.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^5*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^5+10*x^4+25*x^3+15*x^2+x)).

A279639 Exponential transform of the sixth powers A001014.

Original entry on oeis.org

1, 1, 65, 922, 19685, 572036, 16379797, 542459296, 20028938953, 787480005520, 33447797179721, 1522102664036384, 73362723948758125, 3738119667151161280, 200625910519541044189, 11290451562860730241216, 664399657108812332697233, 40781390340823661046136064

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..405

Column k=6 of A279636.

Cf. A001014.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^6*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^6+15*x^5+65*x^4+90*x^3+31*x^2+x)).

A279640 Exponential transform of the seventh powers A001015.

Original entry on oeis.org

1, 1, 129, 2572, 75053, 3228316, 129317797, 6067246144, 321040274585, 17868559819600, 1077981490855241, 69836621117631424, 4779269972787701701, 345917535629669229760, 26385854407691758178669, 2109071203770345994999936, 176264605227324999129677873

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..384

Column k=7 of A279636.

Cf. A001015.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^7*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^7+21*x^6+140*x^5+350*x^4+301*x^3+63*x^2+x)).

A279641 Exponential transform of the eighth powers A001016.

Original entry on oeis.org

1, 1, 257, 7330, 289925, 18565676, 1042651237, 69221777920, 5270005429705, 415374654294352, 35626036180630121, 3293064510986584544, 320276195119275204493, 32969303384902657225792, 3579970600334581051222093, 406942001917387287570455296

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..366

Column k=8 of A279636.

Cf. A001016.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^8*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^8+28*x^7+266*x^6+1050*x^5+1701*x^4+966*x^3 +127*x^2 +x)).

A279642 Exponential transform of the ninth powers A001017.

Original entry on oeis.org

1, 1, 513, 21220, 1130381, 108174916, 8543324917, 800980035472, 88064461381913, 9832425683734288, 1199454069536074601, 158528649288125900224, 21925314644323181005477, 3213026006947537325856832, 497390236613387084643144029, 80481275337746709959509939456

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..348

Column k=9 of A279636.

Cf. A001017.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^9*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^9+36*x^8+462*x^7+2646*x^6+6951*x^5+7770*x^4 +3025*x^3 +255*x^2+x)).

cp's OEIS Frontend

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

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A033462 Exponential (or "EXP") transform of squares A000290.

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PARI

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A279358 Exponential transform of the cubes A000578.

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A279644 a(n) is the n-th term of the exponential transform of the n-th powers.

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A279637 Exponential transform of the fourth powers A000583.

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A279638 Exponential transform of the fifth powers A000584.

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A279639 Exponential transform of the sixth powers A001014.

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