A001373 -id:A001373 - cp's OEIS frontend

A001372 Number of unlabeled mappings (or mapping patterns) from n points to themselves; number of unlabeled endofunctions.

1, 1, 3, 7, 19, 47, 130, 343, 951, 2615, 7318, 20491, 57903, 163898, 466199, 1328993, 3799624, 10884049, 31241170, 89814958, 258604642, 745568756, 2152118306, 6218869389, 17988233052, 52078309200, 150899223268, 437571896993, 1269755237948, 3687025544605, 10712682919341, 31143566495273, 90587953109272, 263627037547365

Offset: 0

N. J. A. Sloane

			The a(3) = 7 mappings are:
1->1, 2->2, 3->3
1->1, 2->2, 3->1 (equiv. to 1->1, 2->2, 3->2, or 1->1, 2->1, 3->3, etc.)
1->1, 2->3, 3->2
1->1, 2->1, 3->2
1->1, 2->1, 3->1
1->2, 2->3, 3->1
1->2, 2->1, 3->1

F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 41, 209.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.401.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 70, Table 3.4.1.
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 = 0..1000 (first 501 terms from Christian G. Bower)
A. L. Agore, A. Chirvasitu, and G. Militaru, The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs, arXiv:2303.06700 [math.QA], 2023.
N. G. de Bruijn, Enumeration of mapping patterns, J. Combin. Theory, 12 (1972), 14-20.
N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 359--368. MR0666861(84i:05008).
Robert L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
Oscar Defrain, Antonio E. Porreca and Ekaterina Timofeeva, Polynomial-delay generation of functional digraphs up to isomorphism, arXiv:2302.13832 [cs.DS], 2023-2024.
Oscar Defrain, Antonio E. Porreca and Ekaterina Timofeeva, Polynomial-delay generation of functional digraphs up to isomorphism, Disc. Appl. Math., vol 357 (2024), pp. 24-33.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 480
Alexander Heaton, Songpon Sriwongsa, and Jeb F. Willenbring, Branching from the General Linear Group to the Symmetric Group and the Principal Embedding, Algebraic Combinatorics, vol. 4, issue 2 (2021), p. 189-200.
Florent Hivert, Jean-Christophe Novelli, and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 144
Sujoy Mukherjee and Petr Vojtěchovský, Minimal representatives of endofunctions, Univ. Denver (2024).
Ronald C. Read, Note on number of functional digraphs, Math. Ann., vol. 143 (1961), pp. 109-111.
Marko Riedel, stackexchange.com, Enumeration of functions by Simultaneous Power Group Enumeration (SPGE)
Marko Riedel, Maple code for sequence using SPGE
Sara Riva, Factorisation of Discrete Dynamical Systems, Ph.D. Thesis, Univ. Côte d'Azur (France 2023).
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Transforms
P. R. Stein, Letter to N. J. A. Sloane, Jun 02 1971

Cf. A000312, A002861, A006961, A001373, A054050, A054745.

with(combstruct): M[ 2671 ] := [ F,{F=Set(K), K=Cycle(T), T=Prod(Z,Set(T))},unlabeled ]:
a:=seq(count(M[2671],size=n),n=0..27); # added by W. Edwin Clark, Nov 23 2010

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]  (* after code given by Robert A. Russell in A000081 *) (* Geoffrey Critzer, Oct 12 2012 *)
max = 40; A[n_] := A[n] = If[n <= 1, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1}]/(n-1)]; H[t_] := Sum[A[n]*t^n, {n, 0, max}]; F = 1 / Product[1 - H[x^n], {n, 1, max}] + O[x]^max; CoefficientList[F, x] (* Jean-François Alcover, Dec 01 2015, after Joerg Arndt *)

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );
A000081=concat([0], A);
H(t)=subst(Ser(A000081, 't), 't, t);
x='x+O('x^N);
F=1/prod(n=1,N, 1 - H(x^n));
Vec(F)
\\ Joerg Arndt, Jul 10 2014

Euler transform of A002861.

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.9557652856519949747148... (Otter's rooted tree constant), c = 0.442876769782206479836... (for a closed form see "Mathematical Constants", p.308). - Vaclav Kotesovec, Mar 17 2015

More terms etc. from Paul Zimmermann, Mar 15 1996

Name edited by Keith J. Bauer, Jan 07 2024

A048888 a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, 10779, 20387, 38673, 73561, 140267, 268065, 513349, 984910, 1892874, 3643569, 7023561, 13557019, 26200181, 50691977, 98182665, 190353369, 369393465, 717457655

Offset: 0

Clark Kimberling

nonn

From Marc LeBrun, Dec 12 2001: (Start)
Define a "numbral arithmetic" by replacing addition with binary bitwise inclusive-OR (so that [3] + [5] = [7] etc.) and multiplication becomes shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc.). [d] divides [n] means there exists an [e] with [d] * [e] = [n]. For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14]. Then it appears that this sequence gives the number of proper divisors of [2^n-1]. Conjecture confirmed by Richard C. Schroeppel, Dec 14 2001. (End)
The number of "prime endofunctions" on n points, meaning the cardinality of the subset of the A001372(n) mappings (or mapping patterns) up to isomorphism from n (unlabeled) points to themselves (endofunctions) which are neither the sum of prime endofunctions (i.e., whose disjoint connected components are prime endofunctions) nor the categorical product of prime endofunctions. The n for which a(n) is prime (n such that the number of prime endofunctions on n points is itself prime) are 2, 4, 5, 6, 9, 13, 19, ... - Jonathan Vos Post, Nov 19 2006
For n>=1, compositions p(1)+p(2)+...+p(m)=n such that p(k)<=p(1)+1, see example. - Joerg Arndt, Dec 28 2012

			From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 2 1 1 1 ]
[ 7]  [ 1 2 1 2 ]
[ 8]  [ 1 2 2 1 ]
[ 9]  [ 2 1 1 1 1 ]
[10]  [ 2 1 1 2 ]
[11]  [ 2 1 2 1 ]
[12]  [ 2 1 3 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 3 1 1 1 ]
[17]  [ 3 1 2 ]
[18]  [ 3 2 1 ]
[19]  [ 3 3 ]
[20]  [ 4 1 1 ]
[21]  [ 4 2 ]
[22]  [ 5 1 ]
[23]  [ 6 ]
(End)

D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.

Cf. A007059.

Cf. A000312, A001372, A002861, A006961, A001373, A054050, A054745, A125024.

N = 66;  x = 'x + O('x^N);
gf = sum(n=0,N,  (1-x^n)*x^n/(1-2*x+x^(n+1)) ) + 'c0;
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Apr 14 2013 */

G.f.: Sum_{k>0} x^k*(1-x^k)/(1-2*x+x^(k+1)). - Vladeta Jovovic, Feb 25 2003

a(m) = Sum_{ n=2..m+1 } Fn(m) where Fn is a Fibonacci n-step number (Fibonacci, tetranacci, etc.) indexed as in A000045, A000073, A000078. - Gerald McGarvey, Sep 25 2004

A329228 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled vertices such that every vertex has outdegree k, n >= 1, 0 <= k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 13, 79, 13, 1, 1, 40, 1499, 1499, 40, 1, 1, 100, 35317, 257290, 35317, 100, 1, 1, 291, 967255, 56150820, 56150820, 967255, 291, 1, 1, 797, 29949217, 14971125930, 111359017198, 14971125930, 29949217, 797, 1

Offset: 1

Andrew Howroyd, Nov 08 2019

			Triangle begins:
  1;
  1,   1;
  1,   2,      1;
  1,   6,      6,        1;
  1,  13,     79,       13,        1;
  1,  40,   1499,     1499,       40,      1;
  1, 100,  35317,   257290,    35317,    100,   1;
  1, 291, 967255, 56150820, 56150820, 967255, 291, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=0..5 are A000012, A001373, A129524, A185193, A185194, A185303.

Row sums are A329234.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
E(v, x) = {my(r=1/(1-x)); for(i=1, #v, r=serconvol(r, prod(j=1, #v, my(g=gcd(v[i], v[j])); (1 + x^(v[j]/g))^g)/(1 + x))); r}
Row(n)={my(s=0); forpart(p=n, s+=permcount(p)*E(p, x+O(x^n))); Vec(s/n!)}
{ for(n=1, 8, print(Row(n))) }

A129524 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 2.

Original entry on oeis.org

0, 0, 1, 6, 79, 1499, 35317, 967255, 29949217, 1033242585, 39323062014, 1637375262965, 74075329383599, 3619112881630497, 189953824713590782, 10661151595417930874, 637230479685691806302, 40415532825383300892418, 2711124591869919503655096

Offset: 1

Vladeta Jovovic, May 29 2007

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
L. Travis, [math/9811127] Graphical Enumeration: A Species-Theoretic Approach

Column k=2 of A329228.

Cf. A001373, A185193, A185194, A185303.

Terms a(10) and beyond from Andrew Howroyd, Nov 08 2019

A185193 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 3.

Original entry on oeis.org

0, 0, 0, 1, 13, 1499, 257290, 56150820, 14971125930, 4829990898461, 1864386642498918, 851204815909786099, 454661054439318678263, 281270600132956104641972, 199701092658236514672384967, 161392692052798327047616107614, 147373164027242947672475065773269

Offset: 1

Nathaniel Johnston, Feb 08 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
L. Travis, [math/9811127] Graphical Enumeration: A Species-Theoretic Approach

Column k=3 of A329228.

Cf. A001373, A129524, A185194, A185303.

Terms a(10) and beyond from Andrew Howroyd, Nov 08 2019

A185194 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 40, 35317, 56150820, 111359017198, 278086517599356, 877760741062694898, 3482578978170418753715, 17204168691253789080138981, 104690934973509839187285618311, 776311587313178356520412354767734, 6942595716239018207126337605515965388

Offset: 1

Nathaniel Johnston, Feb 08 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
L. Travis, [math/9811127] Graphical Enumeration: A Species-Theoretic Approach

Column k=4 of A329228.

Cf. A001373, A129524, A185193, A185303.

Terms a(10) and beyond from Andrew Howroyd, Nov 08 2019

A185303 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 100, 967255, 14971125930, 278086517599356, 6521004095675547914, 197419530111112377546537, 7747427934648623352166753715, 392370903258277676503800999871543, 25436929780226775791085690703723141426, 2090584629532654146005764252197925046719651

Offset: 1

Nathaniel Johnston, Feb 08 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
L. Travis, [math/9811127] Graphical Enumeration: A Species-Theoretic Approach

Column k=5 of A329228.

Cf. A001373, A129524, A185193, A185194.

Terms a(10) and beyond from Andrew Howroyd, Nov 08 2019

A362899 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with k endofunctions.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 9, 6, 1, 1, 0, 1, 22, 162, 13, 1, 1, 0, 1, 63, 3935, 4527, 40, 1, 1, 0, 1, 136, 81015, 1497568, 172335, 100, 1, 1, 0, 1, 302, 1369101, 384069023, 883538845, 7861940, 291, 1, 1, 0, 1, 580, 19601383, 78954264778, 3450709120355, 725601878962, 416446379, 797, 1

Offset: 0

Andrew Howroyd, May 10 2023

Isomorphism is up to permutation of the elements of the n-set. Each endofunction can be considered to be a loopless digraph where each node has out-degree 1.

			Array begins:
==============================================================
n/k| 0  1      2         3             4                 5 ...
---+----------------------------------------------------------
0  | 1  1      1         1             1                 1 ...
1  | 1  0      0         0             0                 0 ...
2  | 1  1      1         1             1                 1 ...
3  | 1  2      9        22            63               136 ...
4  | 1  6    162      3935         81015           1369101 ...
5  | 1 13   4527   1497568     384069023       78954264778 ...
6  | 1 40 172335 883538845 3450709120355 10786100835304758 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..3 are A000012, A001373, A362900, A362901.
Main diagonal is A362902.
Cf. A362644, A362759, A362897.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(v,m) = {prod(i=1, #v, my(g=gcd(v[i],m), e=v[i]/g); (sum(j=1, #v, my(t=v[j]); if(e%(t/gcd(t,m))==0, t)) - 1)^g)}
T(n,k) = {if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(q,m)*x^m/m, O(x*x^k))), k)); s/n!)}

A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

Original entry on oeis.org

0, 1, 12, 153, 2272, 39225, 776736, 17398969, 435538944, 12058401393, 366021568000, 12090393761721, 431832459644928, 16585599200808937, 681703972229640192, 29858718555221585625, 1388451967046195347456, 68316647610168842824161, 3546179063131198669848576, 193670918442059606406896473

Offset: 0

N. J. A. Sloane

This formula is given as a solution to Exercise 1.15a in the Harary and Palmer reference on page 30. However, the formula may not be correct and could be a misprint for Sum_{k=2..n} n! * n^(n-k-1) / (n-k)! which is a formula for A000435(n). - Andrew Howroyd, Feb 06 2024
It appears that a(n) * n^-(n+1) is the mean position of the first duplicate in sequences of n elements randomly drawn with replacement. - Brian P Hawkins, Jan 06 2024
Total count over all mappings from [n] to [n] of tail length plus cycle size of all nodes, where mappings are sets of cycles of trees and tail length is the distance to the cycle that eventually traps the iterates of a node of the mapping; cycle size is the size of that cycle. Alternatively, number of elements on the trajectory of iterates of a node until a repeat is seen, summed over all nodes and mappings. - Marko Riedel, Jul 20 2024

			Example: Consider the map [1,2,3,4] -> [2,3,4,4]. The trajectory of node one is [1,2,3,4]. Hence the tail length is three and the cycle size is one, a fixed point.

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Exercise 1.15a.
P. Flajolet and A. Odlyzko, Random Mapping Statistics, INRIA RR 1114.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marko Riedel, Pusieux series of the tree function and random mapping statistics.

Cf. A000435, A001373, A007778, A262970.

a := proc(n) local k; add(n!*n^(n-k+1)/(n-k)!, k=0..n); end;
# Alternative, e.g.f.:
T := -LambertW(-x): egf := (T + T^2)/(1 - T)^4: ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n = 0..19);  # Peter Luschny, Jul 20 2024

Table[Sum[n!*n^(n-k+1)/(n-k)!, {k, 1, n}], {n, 0, 19}] (* James C. McMahon, Feb 07 2024 *)
a[n_] := n E^n Gamma[n + 1, n] - n^(n + 1);
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Jul 20 2024 *)

a(n) = sum(k=1, n, n! * n^(n-k+1) / (n-k)!) \\ Andrew Howroyd, Jan 06 2024

def a(n):
    total_sum = 0
    for k in range(1, n + 1):
        term = (math.factorial(n) / math.factorial(n - k))*(k**2)*(n**(n - k))
        total_sum += term
    return total_sum
# Brian P Hawkins, Jan 06 2024

a(n) = n^2 * A001865(n). - Gerald McGarvey, Apr 17 2008
a(n) = Sum_{k=1..n} n! * k^2 * n^(n-k) / (n-k)!. - Brian P Hawkins, Jan 06 2024
a(n) = n! * [z^n] (T+T^2)/(1-T)^4 where T is Cayley's tree function T(z) = Sum_{n >= 1} n^(n-1) * z^n/n!. - Marko Riedel, Jul 20 2024
a(n) ~ n^n * ((1/2) * n * sqrt(2 * Pi * n) - (1/3) * n) - Marko Riedel, Jul 20 2024
a(n) = n * e^n * Gamma(n + 1, n) - n^(n + 1) = 2*A262970(n) - A007778(n). - Peter Luschny, Jul 20 2024

Offset corrected by Brian P Hawkins, Jan 06 2024
Name edited by Andrew Howroyd, Feb 06 2024
Offset set to 0 and a(0) = 0 prepended by Marko Riedel, Jul 20 2024

A217897 Triangular array read by rows. T(n,k) is the number of unlabeled functions on n nodes that have exactly k fixed points, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 1, 1, 6, 7, 4, 1, 1, 13, 19, 9, 4, 1, 1, 40, 47, 27, 10, 4, 1, 1, 100, 130, 68, 29, 10, 4, 1, 1, 291, 343, 195, 76, 30, 10, 4, 1, 1, 797, 951, 523, 220, 78, 30, 10, 4, 1, 1, 2273, 2615, 1477, 600, 228, 79, 30, 10, 4, 1, 1, 6389, 7318, 4096, 1708, 625, 230, 79, 30, 10, 4, 1, 1

Offset: 0

Geoffrey Critzer, Oct 14 2012

Row sums are A001372;
  Column for k=0 is A001373;
  Column for k=1 is A001372. (offset)

			Triangle begins:
     1;
     0,    1;
     1,    1,    1;
     2,    3,    1,   1;
     6,    7,    4,   1,   1;
    13,   19,    9,   4,   1,  1;
    40,   47,   27,  10,   4,  1,  1;
   100,  130,   68,  29,  10,  4,  1,  1;
   291,  343,  195,  76,  30, 10,  4,  1, 1;
   797,  951,  523, 220,  78, 30, 10,  4, 1, 1;
  2273, 2615, 1477, 600, 228, 79, 30, 10, 4, 1, 1;

N. J. A. Sloane, Illustration of initial terms of A001372

Needs["Combinatorica`"]; nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];cfd=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,2,30}]],1]; CoefficientList[Series[Product[1/(1-x^i)^cfd[[i]]/(1-y x^i)^rt[[i]],{i,1,nn-1}],{x,0,10}],{x,y}]//Grid (* after code given by Robert A. Russell in A000081 *)

O.g.f.: Product_{n>=1} 1/((1-x^n)^A002862(n) * (1 - y*x^n)^A000081(n) ).

cp's OEIS Frontend

A001372 Number of unlabeled mappings (or mapping patterns) from n points to themselves; number of unlabeled endofunctions.

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A048888 a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.

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A329228 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled vertices such that every vertex has outdegree k, n >= 1, 0 <= k < n.

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A129524 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 2.

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A185193 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 3.

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A185194 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 4.

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A185303 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 5.

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A362899 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with k endofunctions.

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PARI

A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

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