A094573 -id:A094573 - cp's OEIS frontend

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

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).

A060092 Triangle T(n,k) of k-block ordered bicoverings of an unlabeled n-set, n >= 2, k = 3..n+floor(n/2).

Original entry on oeis.org

3, 7, 16, 12, 63, 125, 90, 18, 162, 722, 1716, 1680, 25, 341, 2565, 11350, 27342, 29960, 7560, 33, 636, 7180, 49860, 208302, 503000, 631512, 302400, 42, 1092, 17335, 173745, 1099602, 4389875, 10762299, 14975730, 9632700, 1247400

Offset: 2

Vladeta Jovovic, Feb 26 2001

All columns are polynomials of order binomial(k, 2). - Andrew Howroyd, Jan 30 2020

			[3],
[7, 16],
[12, 63, 125, 90],
[18, 162, 722, 1716, 1680],
[25, 341, 2565, 11350, 27342, 29960, 7560],
[33, 636, 7180, 49860, 208302, 503000, 631512, 302400],
[42, 1092, 17335, 173745, 1099602, 4389875, 10762299, 14975730, 9632700, 1247400], ...
There are 23=7+16 ordered bicoverings of an unlabeled 3-set: 7 3-block bicoverings and 16 4-block bicoverings, cf. A060090.

Andrew Howroyd, Table of n, a(n) for n = 2..1802

Row sums are A060090.
Columns k=3..7 are A055998(n-1), A060091, A060093, A060094, A060095.
Cf. A059443, A060052, A060487, A060492, A094573, A331571.

\\ gives g.f. of k-th column.
ColGf(k) = k!*polcoef(exp(-x - x^2*y/(2*(1-y)) + O(x*x^k))*sum(j=0, k, 1/(1-y)^binomial(j, 2)*x^j/j!), k) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=(3*n\2), y='y + O('y^(n+1))); my(g=serlaplace(exp(-x - x^2*y/(2*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 2)*x^k/k!))); Mat([Col(p/y^2, -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=2, matsize(A)[1], print(A[n, 3..3*n\2])) } \\ Andrew Howroyd, Jan 30 2020

E.g.f. for k-block ordered bicoverings of an unlabeled n-set is exp(-x-x^2/2*y/(1-y))*Sum_{k=0..inf} 1/(1-y)^binomial(k, 2)*x^k/k!.

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

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