A002718 -id:A002718 - cp's OEIS frontend

A059443 Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.

1, 4, 4, 13, 39, 25, 3, 40, 280, 472, 256, 40, 121, 1815, 6185, 7255, 3306, 535, 15, 364, 11284, 70700, 149660, 131876, 51640, 8456, 420, 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105, 3280, 416560, 7894992, 44659776, 103290096

Offset: 2

N. J. A. Sloane, Feb 01 2001

			T(2,3) = 1: 1|12|2.
T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.
T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
Triangle T(n,k) begins:
:    1;
:    4,     4;
:   13,    39,     25,       3;
:   40,   280,    472,     256,      40;
:  121,  1815,   6185,    7255,    3306,     535,     15;
:  364, 11284,  70700,  149660,  131876,   51640,   8456,    420;
: 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;
...

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.

Alois P. Heinz, Rows n = 2..60, flattened
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.]

Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.
Row sums are A002718.
Main diagonal gives A275517.
Right border gives A275521.

nmax = 8; 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]!; row[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3]; Table[ row[n], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Sep 21 2012 *)

\ps 22;
s = 8; pv = vector(s); for(n=1,s,pv[n]=round(polcoeff(f(x,y),n,y)*n!));
for(n=1,s,for(m=3,poldegree(pv[n],x),print1(polcoeff(pv[n],m),", "))) \\ Gerald McGarvey, Dec 03 2009

E.g.f. for m-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).

T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - David Pasino, Sep 22 2016

More terms and additional comments from Vladeta Jovovic, Feb 14 2001

a(37) corrected by Gerald McGarvey, Dec 03 2009

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).

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

Original entry on oeis.org

1, 0, 1, 5, 43, 518, 8186, 163356, 3988342, 116396952, 3985947805, 157783127673, 7131072006829, 364166073164914, 20827961078794845, 1323968417981743817, 92917890994442697487, 7157607311779373890120, 602043767970637640566684

Offset: 0

Vladeta Jovovic, Feb 15 2001

A bicovering is called an r-bicovering if the intersection of every two blocks contains at most one element.
Another name for this sequence is the number of restricted proper 2-covers of [1,...,n].
Number of T_0 2-regular set-systems on an n-set. - Andrew Howroyd, Jan 08 2020

			There are 5 r-bicoverings of a 3-set: 1 3-block bicovering {{1, 2}, {1, 3}, {2, 3}} and 4 4-block bicoverings {{1}, {2}, {3}, {1, 2, 3}}, {{2}, {3}, {1, 2}, {1, 3}}, {{1}, {3}, {1, 2}, {2, 3}}, {{1}, {2}, {1, 3}, {2, 3}}.
G.f. = 1 + x^2 + 5*x^3 + 43*x^4 + 518*x^5 + 8186*x^6 + 163356*x^7 + ...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. (See p. 203.)

Robert Gerbicz, Table of n, a(n) for n = 0..100
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, arXiv:0707.0664 [math.CO], 2007.
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240. (See v_n.)

Row 2 of A331039.
Row sums of A060052.
Cf. A060051, A002718, A059443, A003462, A059945-A059951.

A060053 := proc(n) local h, m; h := (m,n) -> add((-1/2)^k*binomial(m*(m-1)/2,n-k)/k!, k=0..n); n!*add(h(m,n)/m!, m=0..3*n); ceil(evalf(%/exp(1),99)) end: seq(A060053(i), i=0..18);
# Caveat computator! Limited accuracy. Do not use it for n > 50. - Peter Luschny, Jul 06 2011

f[n_] := FullSimplify[(n!/E)*Sum[(1/m!)*Sum[(-1/2)^k*Binomial[m*(m - 1)/2,
n - k]/k!, {k, 0, n}], {m, 0, Infinity}]] (* Robert G. Wilson v, Jul 03 2011 *)

a(n)=round(n!/exp(1)*sum(m=0,3*n+1,1/m!*sum(k=0,n,(-1/2)^k*binomial(m*(m-1)/2,n-k)/k!)))

\\ 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(A=egf1(n), B=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(-x/2 + O(x*x^n))*sum(k=0, n, polcoef(A,k)*B^k)))} \\ Andrew Howroyd, Jan 13 2020

E.g.f. for number of k-block r-bicoverings of an n-set is exp(-x-x^2*y/2)*Sum_{i=0..inf} (1+y)^binomial(i, 2)*x^i/i!.
a(n) = row sums of A060052.
Inverse binomial transform of A014500. - Vladeta Jovovic, Aug 22 2006
The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(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).
E.g.f.: exp(-x/2)*(Sum_{k>=0} A020556(k)*(log(1 + x)/2)^k/k!). - Andrew Howroyd, Jan 13 2020

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 15, 8, 0, 0, 52, 80, 5, 0, 0, 203, 1088, 205, 1, 0, 0, 877, 19232, 11301, 278, 0, 0, 0, 4140, 424400, 904580, 67198, 205, 0, 0, 0, 21147, 11361786, 101173251, 24537905, 250735, 80, 0, 0, 0, 115975, 361058000, 15207243828, 13744869502

Offset: 1

R. H. Hardin, Mar 31 2011

			Array begins:
============================================================================
n\k| 1 2 3   4       5          6             7              8             9
---+------------------------------------------------------------------------
1  | 1 2 5  15      52        203           877           4140         21147
2  | 0 1 8  80    1088      19232        424400       11361786     361058000
3  | 0 0 5 205   11301     904580     101173251    15207243828 2975725761202
4  | 0 0 1 278   67198   24537905   13744869502 11385203921707 ...
5  | 0 0 0 205  250735  425677958 1184910460297 ...
6  | 0 0 0  80  621348 5064948309 ...
7  | 0 0 0  15 1058139 ...
8  | 0 0 0   1 ...
...
Some solutions for 16 X 4:
  1 1 1 0    1 1 1 1    1 1 1 1    1 1 1 0    1 1 1 1
  1 0 1 1    1 1 0 1    1 1 0 0    1 0 1 1    1 1 0 0
  1 0 1 0    1 0 1 1    1 0 1 1    1 0 0 1    1 0 1 1
  1 0 0 1    1 0 0 0    1 0 0 0    1 0 0 0    1 0 0 0
  0 1 1 1    0 1 1 0    0 1 1 1    0 1 1 0    0 1 1 1
  0 1 0 1    0 1 0 0    0 1 0 0    0 1 0 1    0 1 0 0
  0 1 0 0    0 0 1 1    0 0 1 1    0 1 0 0    0 0 1 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 1 1    0 0 0 1
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0

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

Rows 1..5 are A000110, A002718, A060486, A188446, A188447.
Columns 5..6 are A331127, A331129.
Column sums are A319190.
Cf. A059443, A060487, A188392, A219585, A318361, A330942, A330964, A331039.

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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018

A(n,k) = 0 for n > 2^(k-1). - Andrew Howroyd, Jan 24 2020

A020554 Number of multigraphs on n labeled edges (without loops).

Original entry on oeis.org

1, 1, 3, 16, 139, 1750, 29388, 624889, 16255738, 504717929, 18353177160, 769917601384, 36803030137203, 1984024379014193, 119571835094300406, 7995677265437541258, 589356399302126773920, 47609742627231823142029, 4193665147256300117666879

Offset: 0

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

Or, number of bicoverings of an n-set.
Or, number of 2-covers of [1,...,n].
Also the number of set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
The a(3) = 16 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123) (2)(13)(123) (3)(12)(123) (12)(13)(23)
  (1)(1)(23)(23) (1)(2)(3)(123) (1)(2)(13)(23) (1)(3)(12)(23) (2)(2)(13)(13) (2)(3)(12)(13) (3)(3)(12)(12)
  (1)(1)(2)(3)(23) (1)(2)(2)(3)(13) (1)(2)(3)(3)(12)
  (1)(1)(2)(2)(3)(3)
(End)

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
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see s_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.]
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 2 of A188392.

Cf. A002718, A007716, A020555, A050535, A094574, A316974.

Ceiling[ CoefficientList[ Series[ Exp[ -1 + (Exp[ z ] - 1)/2 ]Sum[ Exp[ s(s - 1)z/2 ]/s!, {s, 0, 21} ], {z, 0, 9} ], z ] Table[ n!, {n, 0, 9} ] ] (* Mitch Harris, May 01 2004 *)
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[Select[mps[Ceiling[Range[1/2,n,1/2]]],And@@UnsameQ@@@#&]],{n,5}] (* Gus Wiseman, Jul 18 2018 *)

E.g.f.: exp(-3/2+exp(x)/2) * Sum_{n>=0} exp(binomial(n, 2)*x)/n! [Comtet]. - Vladeta Jovovic, Apr 27 2004
E.g.f. (an equivalent version in Maple format): G:=exp(-1+(exp(z)-1)/2)*sum(exp(s*(s-1)*z/2)/s!, s=0..infinity);
E.g.f.: exp((exp(x)-1)/2)*Sum_{n>=0} A020556(n)*(x/2)^n/n!. - Vladeta Jovovic, May 02 2004
Stirling_2 transform of A014500.
The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).

A060090 Number of ordered bicoverings of an unlabeled n-set.

Original entry on oeis.org

1, 0, 3, 23, 290, 4298, 79143, 1702923, 42299820, 1188147639, 37276597020, 1291633545897, 48995506718702, 2019395409175529, 89864601931874318, 4294295828157319651, 219321170795303112118, 11922219151375200468886

Offset: 0

Vladeta Jovovic, Feb 25 2001

nonn

			There are 23 ordered bicoverings of an unlabeled 3-set, 7 3-block bicoverings:
1 ( { 3 }, { 1, 2 }, { 1, 2, 3 } )
2 ( { 3 }, { 1, 2, 3 }, { 1, 2 } )
3 ( { 2, 3 }, { 1 }, { 1, 2, 3 } )
4 ( { 2, 3 }, { 1, 3 }, { 1, 2 } )
5 ( { 2, 3 }, { 1, 2, 3 }, { 1 } )
6 ( { 1, 2, 3 }, { 3 }, { 1, 2 } )
7 ( { 1, 2, 3 }, { 2, 3 }, { 1 } )
and 16 4-block bicoverings:
1 ( { 3 }, { 2 }, { 1 }, { 1, 2, 3 } )
2 ( { 3 }, { 2 }, { 1, 3 }, { 1, 2 } )
3 ( { 3 }, { 2 }, { 1, 2 }, { 1, 3 } )
4 ( { 3 }, { 2 }, { 1, 2, 3 }, { 1 } )
5 ( { 3 }, { 2, 3 }, { 1 }, { 1, 2 } )
6 ( { 3 }, { 2, 3 }, { 1, 2 }, { 1 } )
7 ( { 3 }, { 1, 2 }, { 2 }, { 1, 3 } )
8 ( { 3 }, { 1, 2 }, { 2, 3 }, { 1 } )
9 ( { 3 }, { 1, 2, 3 }, { 2 }, { 1 } )
10 ( { 2, 3 }, { 3 }, { 1 }, { 1, 2 } )
11 ( { 2, 3 }, { 3 }, { 1, 2 }, { 1 } )
12 ( { 2, 3 }, { 1 }, { 3 }, { 1, 2 } )
13 ( { 2, 3 }, { 1 }, { 1, 3 }, { 2 } )
14 ( { 2, 3 }, { 1, 3 }, { 2 }, { 1 } )
15 ( { 2, 3 }, { 1, 3 }, { 1 }, { 2 } )
16 ( { 1, 2, 3 }, { 3 }, { 2 }, { 1 } )

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=2 of A331571.
Row sums of A060092.
Cf. A060069, A060070, A060051, A060052, A060053, A002718, A059443.

seq(n)={my(m=3*n\2, y='y + O('y^(n+1))); Vec(subst(Pol(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!))), x, 1))} \\ Andrew Howroyd, Jan 30 2020

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

A060051 Number of n-block r-bicoverings.

Original entry on oeis.org

1, 0, 0, 2, 79, 82117, 4936900199, 27555467226181396, 20554872166566046969648895, 2786548447182420815380482508924733911, 89607283195144164483079065133414172790220498449945, 864608448649084311874549352448884076627916391005243593208944730790

Offset: 0

Vladeta Jovovic, Feb 15 2001

A bicovering is an r-bicovering if the intersection of every two blocks contains at most one element.

			There are 2 3-block r-bicoverings: {{1},{2},{1,2}} and {{1,2},{1,3},{2,3}}.

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

Andrew Howroyd, Table of n, a(n) for n = 0..30

Column sums of A060052.

Cf. A002718, A060053, A060069, A060070.

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

Terms a(11) and beyond from Andrew Howroyd, Jan 30 2020

A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).

Original entry on oeis.org

1, 3, 1, 7, 57, 95, 43, 3, 35, 717, 3107, 4520, 2465, 445, 12, 155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70, 651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

			Triangle begins:
  [1, 3, 1];
  [7, 57, 95, 43, 3];
  [35, 717, 3107, 4520, 2465, 445, 12];
  [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70];
  [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465];
   ...
There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.

Andrew Howroyd, Table of n, a(n) for n = 3..1157

Columns include A060483, A060484, A060485.
Row sums are A060486.
Cf. A006095, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

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]!)}
row(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])*y^(m-n)/(1+y))}
for(n=3, 8, print(Vecrev(row(3,n)))); \\ Andrew Howroyd, Dec 23 2018

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

A059945 Number of 4-block bicoverings of an n-set.

Original entry on oeis.org

0, 0, 4, 39, 280, 1815, 11284, 68859, 416560, 2509455, 15086764, 90610179, 543928840, 3264374295, 19588645444, 117539063499, 705255937120, 4231600258335, 25389795391324, 152339353740819, 914037866361400, 5484232429393575, 32905410268988404, 197432508689714139

Offset: 1

Vladeta Jovovic, Feb 14 2001

			There are 4 4-block bicoverings of a 3-set: {{1},{2},{3},{1,2,3}}, {{2},{3},{1,2},{1,3}}, {{1},{3},{1,2},{2,3}} and {{1},{2},{1,3},{2,3}}.

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

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Column k=4 of A059443.

Cf. A002718.

With[{c=1/4!},Table[c(6^n-4 3^n-3 2^n+12),{n,20}]] (* or *) LinearRecurrence[ {12,-47,72,-36},{0,0,4,39},20] (* Harvey P. Dale, Aug 10 2011 *)

a(n) = {(1/4!)*(6^n - 4*3^n - 3*2^n + 12)} \\ Andrew Howroyd, Jan 29 2020

a(n) = (1/4!)*(6^n - 4*3^n - 3*2^n + 12).
E.g.f. for m-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).
a(n) = 12*a(n-1) - 47*a(n-2) + 72*a(n-3) - 36*a(n-4) for n > 4. - Harvey P. Dale, Aug 10 2011
G.f.: -x^3*(9*x-4) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Jan 11 2013

More terms from Colin Barker, Jan 11 2013

A059951 Number of 10-block bicoverings of an n-set.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 420, 154637, 20368816, 1775801814, 124151410020, 7596257673279, 426319554841752, 22564352299016528, 1146221298547133380, 56531610963314602401, 2728475248127447671008, 129586638359127411410442, 6080467290450346517206500, 282689089820505452872162403

Offset: 1

Vladeta Jovovic, Feb 14 2001

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

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column k=10 of A059443.

Cf. A002718.

a(n) = (1/10!)*(45^n - 10*36^n - 45*29^n + 90*28^n + 360*22^n - 480*21^n + 630*17^n - 2520*16^n + 2100*15^n - 3780*12^n + 10080*11^n - 6552*10^n - 3150*9^n + 18900*8^n - 31500*7^n + 28560*6^n - 46620*5^n + 27720*4^n + 85560*3^n - 146160*2^n + 83520).
E.g.f. for m-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).
G.f.: -x^7*(5467233152463667200*x^14 -6460773223081605120*x^13 +3312489509664336576*x^12 -965946275708647680*x^11 +175045400422088532*x^10 -19853467917718628*x^9 +1255863452001343*x^8 -11591551437545*x^7 -5424120630669*x^6 +520759916751*x^5 -24697320639*x^4 +659527325*x^3 -8843563*x^2 +25697*x +420) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(12*x -1)*(15*x -1)*(16*x -1)*(17*x -1)*(21*x -1)*(22*x -1)*(28*x -1)*(29*x -1)*(36*x -1)*(45*x -1)). - Colin Barker, Jul 09 2013

More terms from Colin Barker, Jul 09 2013

cp's OEIS Frontend

A059443 Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A020554 Number of multigraphs on n labeled edges (without loops).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A060090 Number of ordered bicoverings of an unlabeled n-set.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A060051 Number of n-block r-bicoverings.

Views

Author

Keywords

Comments

Examples

References

Links