A020554 -id:A020554 - cp's OEIS frontend

A002719 Erroneous version of A020554.

Original entry on oeis.org

1, 3, 16, 139, 1750, 29388, 623909

Offset: 1

N. J. A. Sloane, Oct 20 2015

dead

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).

L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

A020556 Number of oriented multigraphs on n labeled arcs (without loops).

Original entry on oeis.org

1, 1, 7, 87, 1657, 43833, 1515903, 65766991, 3473600465, 218310229201, 16035686850327, 1356791248984295, 130660110400259849, 14177605780945123273, 1718558016836289502159, 230999008481288064430879, 34208659263890939390952225, 5549763869122023099520756513

Offset: 0

Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe

nonn

Generalized Bell numbers: a(n) = Sum_{k=2..2*n} A078739(n,k), n >= 1.
Let B_{m}(x) = Sum_{j>=0} exp(j!/(j-m)!*x-1)/j! then
a(n) = n! [x^n] taylor(B_{2}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{2}(x). a(n) is row 2 of the square array representation of A090210. - Peter Luschny, Mar 27 2011
Also the number of set partitions of {1,2,...,2n+1} such that the block |n+1| is a part but no block |m| with m < n+1. - Peter Luschny, Apr 03 2011

			Example: For n = 2 the a(2) = 7 are the number of set partitions of 5 such that the block |3| is a part but no block |m| with m < 3: 3|1245, 3|4|125, 3|5|124, 3|12|45, 3|14|25, 3|15|24, 3|4|5|12. - _Peter Luschny_, Apr 05 2011

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Alois P. Heinz, Table of n, a(n) for n = 0..288
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
Peter Luschny, Set partitions
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
M. Riedel, Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition, Mathematics Stack Exchange.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A020554, A014500, A020558, A090210, A095149, A106436, A182930.

A020556 := proc(n) local k;
add((-1)^(n+k)*binomial(n,k)*combinat[bell](n+k),k=0..n) end:
seq(A020556(n),n=0..17); # Peter Luschny, Mar 27 2011
# Uses floating point arithmetic, increase working precision for large n.
A020556 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(3,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*2^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A020556(n),n=0..15); # Peter Luschny, Mar 30 2011
T := proc(n, k) option remember;
  if n = 1 then 1
elif n = k then T(n-1,1) - T(n-1,n-1)
else T(n-1,k) + T(n, k+1) fi end:
A020556 := n -> T(2*n+1,n+1);
seq(A020556(n), n = 0..99); # Peter Luschny, Apr 03 2011

f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]
(* Second program: *)
a[n_] := Sum[(-1)^k*Binomial[n, k]*BellB[2n-k], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 11 2017, after Vladeta Jovovic *)

a(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(k=0, n, (-1)^k*binomial(n,k)*polcoef(bell, 2*n-k))} \\ Andrew Howroyd, Jan 13 2020

a(n) = e*Sum_{k>=0} ((k+2)!^n/(k+2)!)*(k!^n), n>=1.
a(n) = (1/e)*Sum_{k>=2} (k*(k-1))^n/k!, n >= 1. a(0) := 1. (From eq.(26) with r=2 of the Schork reference.)
E.g.f.: (1/e)*(2 + Sum_{k>=2} ((exp(k*(k-1)*x))/k!)) (from top of p. 4656 of the Schork reference).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - Vladeta Jovovic, May 02 2004
a(n) = A095149(2n,n). - Alois P. Heinz, Dec 20 2018
a(n) = A106436(2n,n) = A182930(2n+1,n+1). - Alois P. Heinz, Jan 29 2019

Edited by Robert G. Wilson v, Apr 30 2002

A020555 Number of multigraphs on n labeled edges (with loops). Also number of genetically distinct states amongst n individuals.

Original entry on oeis.org

1, 2, 9, 66, 712, 10457, 198091, 4659138, 132315780, 4441561814, 173290498279, 7751828612725, 393110572846777, 22385579339430539, 1419799938299929267, 99593312799819072788, 7678949893962472351181, 647265784993486603555551, 59357523410046023899154274

Offset: 0

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe, N. J. A. Sloane

nonn

Also the number of factorizations of (p_n#)^2. - David W. Wilson, Apr 30 2001
Also the number of multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
a(n) gives the number of genetically distinct states for n diploid individuals in the case that maternal and paternal alleles transmitted to the individuals are not distinguished (if maternal and paternal alleles are distinguished, then the number of states is A000110(2n)). - Noah A Rosenberg, Aug 23 2022

			From _Gus Wiseman_, Jul 18 2018: (Start)
The a(2) = 9 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (2)(112), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12), (2)(2)(11),
  (1)(1)(2)(2).
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
E. Keith Lloyd, Math. Proc. Camb. Phil. Soc., vol. 103 (1988), 277-284.
A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Alois P. Heinz, Table of n, a(n) for n = 0..310 (first 101 terms from Vincenzo Librandi)
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
Marko Riedel et al., Set partitions of {1,1,2,2,...,n,n}
E. A. Thompson, Gene identities and multiple relationships. Biometrics 30 (1974), 667-680. See Table 5.

Row n=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. A007716, A007717, A014500, A014501, A020554, A094574, A316974.
See also A322764. Row 0 of the array in A322765.
Main diagonal of A346500.
Cf. A001055, A002110.

B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end;
r:=m->[seq(P(m,n),n=0..20)]; r(0); # N. J. A. Sloane, Dec 30 2018

max = 16; s = Series[Exp[-3/2 + Exp[x]/2]*Sum[Exp[Binomial[n+1, 2]*x]/n!, {n, 0, 3*max }], {x, 0, max}] // Normal; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n] // Round, {n, 0, max} ] (* Jean-François Alcover, Apr 23 2014, after Vladeta Jovovic *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[mps[Ceiling[Range[1/2,n,1/2]]]],{n,5}] (* Gus Wiseman, Jul 18 2018 *)

Lloyd's article gives a complicated explicit formula.
E.g.f.: exp(-3/2 + exp(x)/2)*Sum_{n>=0} exp(binomial(n+1, 2)*x)/n! [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
a(n) = A001055(A002110(n)^2). - Alois P. Heinz, Aug 23 2022

A002718 Number of bicoverings of an n-set.

Original entry on oeis.org

1, 0, 1, 8, 80, 1088, 19232, 424400, 11361786, 361058000, 13386003873, 570886397340, 27681861184474, 1511143062540976, 92091641176725504, 6219762391554815200, 462595509951068027741, 37676170944802047077248, 3343539821715571537772071, 321874499078487207168905840

Offset: 0

N. J. A. Sloane

Another description: number of proper 2-covers of [1,...,n].

			For n=3, there are 8 collections of distinct subsets of {1,2,3} with the property that each of 1, 2, and 3 appears in exactly two subsets:
  {1,2,3},{1,2},{3}
  {1,2,3},{1,3},{2}
  {1,2,3},{2,3},{1}
  {1,2,3},{1},{2},{3}
  {1,2},{1,3},{2,3}
  {1,2},{1,3},{2},{3}
  {1,2},{2,3},{1},{3}
  {1,3},{2,3},{1},{2}
Therefore a(3) = 8. - _Michael B. Porter_, Jul 16 2016

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
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..100
Peter Cameron, Thomas Prellberg, Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see t_n).
L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

Cf. A020554, A002719, A003462, A059945-A059951, A060053. Row sums of A059443.

nmax = 16; imax = 2*(nmax - 2); egf := E^(-x - 1/2*x^2*(E^y - 1))*Sum[(x^i/i!)*E^(Binomial[i, 2]*y), {i, 0, imax}]; fx = CoefficientList[Series[egf, {y, 0, imax}], y]*Range[0, imax]!; a[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3] // Total; Table[ a[n] , {n, 2, nmax}] (* Jean-François Alcover, Apr 04 2013 *)

E.g.f. for k-block bicoverings of an n-set is exp(-x-1/2*x^2*(exp(y)-1))*Sum_{i=0..inf} x^i/i!*exp(binomial(i, 2)*y).
Stirling_2 transform of A060053.
The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(e^x-1).
a(n) = Sum_{m=0..n + floor(n/2); k=0..n; s=0..min(m/2,k); t=0..m-2s} Stirling2(n,k) * k!/m! * binomial(m,2s) * A001147(s) * (-1)^(m+s+t) * binomial(m-2s,t) * binomial(t*(t-1)/2,k-s). Interpret m as the number of blocks in a bicovering, k the number of clumps of points that are always all together in blocks. This formula counts bicoverings by quotienting them to the clumpless case (an operation which preserves degree) and counting incidence matrices of those, and counts those matrices as the transposes of incidence matrices of labeled graphs with no isolated points and no isolated edges. - David Pasino, Jul 09 2016

More terms from Vladeta Jovovic, Feb 18 2001

a(0), a(1) prepended by Alois P. Heinz, Jul 29 2016

A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796, 746244486452472386213939, 67796741482683128375533560

Offset: 0

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Alois P. Heinz, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter Cameron, Thomas Prellberg, Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see u_n).
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Row n=2 of A331126.

Cf. A020554, A020555, A014501, A060053.

read("transforms") ;
A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
A014500 := proc(n) local i,gexp,lexp;
gexp := [seq(1/2^i/i!,i=0..n+1)] ;
lexp := add( A020556(i)*((log(1+x))/2)^i/i!,i=0..n+1) ;
lexp := taylor(lexp,x=0,n+1) ;
lexp := gfun[seriestolist](lexp,'ogf') ;
CONV(gexp,lexp) ; op(n+1,%)*n! ; end proc:
seq(A014500(n),n=0..20) ; # R. J. Mathar, Jul 03 2011

max = 20; A020556[n_] := Sum[(-1)^(n+k)*Binomial[n, k]*BellB[n+k], {k, 0, n}]; egf = Exp[x/2]*Sum[A020556[n]*(Log[1+x]/2)^n/n!, {n, 0, max}] + O[x]^max; CoefficientList[egf, x]*Range[0, max-1]! (* Jean-François Alcover, Feb 19 2017, after Vladeta Jovovic *)

\\ here egf1 is A020556 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
seq(n)={my(B=egf1(n), L=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(x/2 + O(x*x^n))*sum(k=0, n, polcoef(B ,k)*L^k)))} \\ Andrew Howroyd, Jan 13 2020

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
E.g.f.: exp(x/2)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004
Binomial transform of A060053.
The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).
The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).

A094574 Number of (<=2)-covers of an n-set.

Original entry on oeis.org

1, 1, 5, 40, 457, 6995, 136771, 3299218, 95668354, 3268445951, 129468914524, 5868774803537, 301122189141524, 17327463910351045, 1109375488487304027, 78484513540137938209, 6098627708074641312182, 517736625823888411991202, 47791900951140948275632148

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

nonn

Also the number of strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. For example, the a(2) = 5 strict multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (11)(22), (1)(2)(12). - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Sep 02 2019: (Start)
These are set-systems covering {1..n} with vertex-degrees <= 2. For example, the a(3) = 40 covers are:
  {123}  {1}{23}    {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}    {1}{2}{13}    {1}{2}{3}{13}
         {3}{12}    {1}{2}{23}    {1}{2}{3}{23}
         {1}{123}   {1}{3}{12}    {1}{2}{13}{23}
         {12}{13}   {1}{3}{23}    {1}{2}{3}{123}
         {12}{23}   {2}{3}{12}    {1}{3}{12}{23}
         {13}{23}   {2}{3}{13}    {2}{3}{12}{13}
         {2}{123}   {1}{12}{23}
         {3}{123}   {1}{13}{23}
         {12}{123}  {1}{2}{123}
         {13}{123}  {1}{3}{123}
         {23}{123}  {2}{12}{13}
                    {2}{13}{23}
                    {2}{3}{123}
                    {3}{12}{13}
                    {3}{12}{23}
                    {12}{13}{23}
                    {1}{23}{123}
                    {2}{13}{123}
                    {3}{12}{123}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A219585. - Alois P. Heinz, Nov 23 2012
Dominated by A003465.
Graphs with vertex-degrees <= 2 are A136281.
Cf. A002718, A007716, A020554, A020555, A050535, A094574, A136284, A316974, A327104, A327106, A327229.
Main diagonal of A346517.

facs[n_]:=facs[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[Array[Prime,n,1,Times]^2],UnsameQ@@#&]],{n,0,6}] (* Gus Wiseman, Jul 18 2018 *)
m = 20;
a094577[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
CoefficientList[egf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 13 2019 *)

Row sums of A094573.

E.g.f: exp(-1-1/2*(exp(x)-1))*Sum(exp(x*binomial(n+1, 2))/n!, n=0..infinity) or exp((1-exp(x))/2)*Sum(A094577 (n)*(x/2)^n/n!, n=0..infinity).

A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 15, 16, 4, 1, 52, 139, 39, 5, 1, 203, 1750, 862, 81, 6, 1, 877, 29388, 35775, 4079, 150, 7, 1, 4140, 624889, 2406208, 507549, 15791, 256, 8, 1, 21147, 16255738, 238773109, 127126912, 5442547, 52450, 410, 9, 1, 115975, 504717929, 32867762616

Offset: 1

R. H. Hardin, Mar 30 2011

			Array begins:
========================================================================
n\k| 1  2   3      4         5            6            7               8
---+--------------------------------------------------------------------
1  | 1  2   5     15        52          203           877           4140
2  | 1  3  16    139      1750        29388        624889       16255738
3  | 1  4  39    862     35775      2406208     238773109    32867762616
4  | 1  5  81   4079    507549    127126912   55643064708 38715666455777
5  | 1  6 150  15791   5442547   4762077620 8738543204786
6  | 1  7 256  52450  46757209 135029200594
7  | 1  8 410 154279 335279744
8  | 1  9 625 411180
9  | 1 10 915
     ...
All solutions for 6 X 2
..1..1....1..1....1..0....1..1
..1..1....1..1....1..0....1..0
..1..0....1..1....1..0....1..0
..0..1....0..0....0..1....0..1
..0..0....0..0....0..1....0..1
..0..0....0..0....0..1....0..0

Andrew Howroyd, Table of n, a(n) for n = 1..181 (terms 1..69 from R. H. Hardin)

Rows 1..8 are A000110, A020554, A165434, A165435, A165436, A165437, A188393, A188394.
Columns 3..7 are A011863(n+1), A175707, A188389, A188390, A188391.
Main diagonal gives A188388.
Cf. A188445, A219727, A330942.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 12 2018

A316974 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 1, 4, 14, 49, 173, 652, 2494

Offset: 0

Gus Wiseman, Jul 17 2018

Also the number of unlabeled multigraphs with n edges, allowing loops, spanning an initial interval of positive integers with no equivalent vertices (two vertices are equivalent if in every edge the multiplicity of the first is equal to the multiplicity of the second). For example, non-isomorphic representatives of the a(2) = 4 multigraphs are {(1,2),(1,3)}, {(1,1),(1,2)}, {(1,1),(2,2)}, {(1,1),(1,1)}.

			Non-isomorphic representatives of the a(3) = 14 strict multiset partitions:
  (112233),
  (1)(12233), (11)(2233), (12)(1233), (112)(233),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (2)(11)(233), (11)(22)(33), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(12)(33), (1)(2)(13)(23).

Cf. A001055, A007716, A007717, A007719, A020554, A020555, A045778, A050535, A053419, A061742, A094574, A162247, A316892, A316972.

a(7) from Andrew Howroyd, Feb 07 2020

A165434 Number of tri-coverings of a set.

Original entry on oeis.org

1, 1, 4, 39, 862, 35775, 2406208, 238773109, 32867762616, 6009498859909, 1412846181645855, 416415343791239162, 150747204270574506888, 65905473934553360340713, 34305461329980340135062217, 21003556204331356488142290707, 14967168378184553824642693791437

Offset: 0

Doron Zeilberger, Sep 18 2009

nonn

			For n=2, a(2)=4, since if you have two sets of identical triples the A-brothers and the B-sisters, and you want to arrange them into a multiset of nonempty sets, where no one is allowed to cohabitate with his or her sibling, the following are possible 1.{{AB},{AB},{AB}} 2.{{AB},{AB},{A},{B}} 3.{{AB},{A},{A},{B},{B}} 4.{{A},{A},{A},{B},{B},{B}}.

Andrew Howroyd, Table of n, a(n) for n = 0..100
E. A. Bender, Partitions of multisets, Discrete Mathematics 9 (1974) 301-312.
J. S. Devitt and D. M. Jackson, The enumeration of covers of a finite set, J. London Math. Soc.(2) 25 (1982), 1-6.
Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls?, has links to more terms and related sequences
Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls?, arXiv:0909.3453 [math.CO], 2009.
Doron Zeilberger, BABUSHKAS; Local copy

Row 3 of A188392.

Cf. A000110 (unicoverings), A020554 (bicoverings).

Do SeqBrn(3,n); in the Maple package BABUSHKAS (see links) where n+1 is the number of desired terms.

Edited by Charles R Greathouse IV, Oct 28 2009

A014501 Number of graphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 2, 7, 43, 403, 5245, 89132, 1898630, 49209846, 1517275859, 54669946851, 2269075206395, 107199678164289, 5707320919486026, 339510756324234931, 22400182888853554291, 1628654713107465602783, 129754625253841669625051

Offset: 0

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Row n=2 of A331161.

Cf. A020554, A020555, A014500.

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n+1, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004

cp's OEIS Frontend

A002719 Erroneous version of A020554.

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A020556 Number of oriented multigraphs on n labeled arcs (without loops).

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A020555 Number of multigraphs on n labeled edges (with loops). Also number of genetically distinct states amongst n individuals.

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A002718 Number of bicoverings of an n-set.

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A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

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A094574 Number of (<=2)-covers of an n-set.

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A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

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