A060052 -id:A060052 - 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

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

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

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!.

A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

Original entry on oeis.org

4, 60, 120, 13, 375, 3030, 9030, 5040, 28, 1392, 24552, 207900, 838320, 1345680, 362880, 50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600, 80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640

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.

All columns are polynomials of order binomial(k, 3). - Andrew Howroyd, Jan 30 2020

			Triangle begins:
  [4, 60, 120],
  [13, 375, 3030, 9030, 5040],
  [28, 1392, 24552, 207900, 838320, 1345680, 362880],
  [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
   ...
There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).

Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)

Row sums are A060491.
Columns k=4..6 are A060488, A060489, A060490.
Cf. A059443, A059530, A060052, A060092, A060487, A331571.

\\ gives g.f. of k-th column.
ColGf(k) = k!*polcoef(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^k) )*sum(j=0, k, 1/(1-y)^binomial(j, 3)*exp((-x^2/2)/(1-y)^j + O(x*x^k))*x^j/j!), k) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))); Mat([Col(p/y^3, -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ Andrew Howroyd, Jan 30 2020

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

A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 39, 89, 43, 3, 0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12, 0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70, 0, 0, 0, 0, 0, 5040, 536760, 6052730, 20660055, 29432319, 19826737, 6481160, 964495, 52430

Offset: 3

Vladeta Jovovic, Feb 22 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.

			Triangle begins:
  [0, 0, 0, 0, 1, 3, 1],
  [0, 0, 0, 0, 1, 39, 89, 43, 3],
  [0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12],
  [0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70],
   ...
There are 5 = 1 + 3 + 1 T_0-tricoverings of a 3-set and 175 = 1 + 39 + 89 + 43 + 3 T_0-tricoverings of a 4-set, cf. A060070.

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 = 3..1674
Vladeta Jovovic, T_0-tricoverings of a 4-set

Column sums are A060069.
Row sums are A060070.
Cf. A059443, A060052, A060092, A060487, A060492, A331039.

\\ gets k-th column as vector
C(k)=if(k<4, [], Vecrev(serlaplace(polcoef(exp(-x + x^2/2 + x^3*y/3 + O(x*x^k))*sum(i=0, 2*k, (1+y)^binomial(i, 3)*exp(-x^2*(1+y)^i/2 + O(x*x^k))*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020

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

E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.

T(n,k) = 0 for n > binomial(k, 3). - Andrew Howroyd, Jan 30 2020

A060491 Number of ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

1, 0, 0, 184, 17488, 2780752, 689187720, 236477490418, 107317805999204, 62318195302890305, 45081693413563797127, 39762626850034005271588, 42009504510315968282400843, 52381340312720286113688037624, 76118747309505733406576769607755

Offset: 0

Vladeta Jovovic, Mar 20 2001

nonn

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

			There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060492).

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A331571.
Row sums of A060492.
Cf. A060486, A060487, A060090, A060092, A060069, A060070, A060051, A060052, A060053, A002718, A059443.

seq(n)={my(m=2*n\2, y='y + O('y^(n+1))); Vec(subst(Pol(serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))), x, 1))} \\ Andrew Howroyd, Jan 30 2020

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

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

A094573 Triangle T(n,k) giving number of (<=2)-covers of an n-set with k blocks.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 12, 20, 7, 1, 39, 169, 186, 59, 3, 1, 120, 1160, 2755, 2243, 661, 55, 1, 363, 7381, 33270, 52060, 33604, 9167, 910, 15, 1, 1092, 45500, 367087, 988750, 1126874, 601262, 151726, 16401, 525, 1, 3279, 276529, 3873786, 17005149

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

Cover of a set is (<=2)-cover if every element of the set is covered with at most two blocks of the cover.

			Triangle T(n,k) begins:
  1;
  1;
  1,   3,    1;
  1,  12,   20,    7;
  1,  39,  169,  186,   59,   3;
  1, 120, 1160, 2755, 2243, 661, 55;
  ...

Row sums give A094574.

Cf. A059443, A060052.

rows = 9; m = rows + 2;
egf = Exp[-x - (x^2/2)*(Exp[y]-1)]*Sum[Exp[y*Binomial[n+1, 2]]*(x^n/n!), {n, 0, m}];
cc = CoefficientList[# + O[x]^m, x]& /@ CoefficientList[egf + O[y]^m, y];
(Range[0, Length[cc]-1]! * cc)[[1 ;; rows]] /. {0, a__} :> {a} // Flatten (* Jean-François Alcover, May 13 2019 *)

E.g.f.: exp(-x-x^2/2*(exp(y)-1))*(Sum_{n>=0} exp(y*binomial(n+1, 2))*x^n/n!).

A276640 Triangle T(n, k) = the number of point-labeled graphs with n points and k edges, no points isolated, no edges isolated. By rows, 0 <= n, ceiling(2*n/3) <= k <= binomial(n, 2).

Original entry on oeis.org

1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1, 90, 1356, 3670, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 1680, 18942, 69450, 156870, 258160, 331506, 343140, 290745, 202755, 116175, 54257, 20349, 5985, 1330, 210, 21, 1

Offset: 1

David Pasino, Sep 08 2016

In an incidence matrix for a graph of this kind, with n columns and k rows, each row has 2 ones (since it is a graph), the rows are distinct (since it is not a multigraph), no column is all zeros (since there are no isolated points), and the columns are distinct (since there are no isolated edges). The transpose of such a matrix, and only such, is an incidence matrix of a covering of a set of k elements (called points) by n distinct nonempty subsets (called blocks) such that every point belongs to exactly 2 blocks, and every 2 blocks have at most 1 point of intersection (for if 2 points each belong to both of 2 blocks, then those 2 blocks are all the blocks that either of those 2 points belong to, so the columns for those 2 points in the matrix are equal). Referring all these matrices to canonical ordered sets of n and k points, the number of matrices for each covering by blocks of these kinds is the factorial of the number of blocks. (Since the rows are distinct, every permutation of the blocks as row indices gives a different matrix.) Hence the number of these graphs, with k blocks on n points, T(n, k), is related to the number of those covers, A060052, by T(n, k) * k! = A060052(k, n) * n!.

			The triangle T(n, k) begins:
n\k 0 1 2 3  4   5    6    7    8    9   10   11   12  13  14
0   1 0 0 0  0   0    0    0    0    0    0    0    0   0   0
1   0 0 0 0  0   0    0    0    0    0    0    0    0   0   0
2   0 0 0 0  0   0    0    0    0    0    0    0    0   0   0
3   0 0 3 1  0   0    0    0    0    0    0    0    0   0   0
4   0 0 0 16 15  6    1    0    0    0    0    0    0   0   0
5   0 0 0 0  125 222  205  120  45   10   1    0    0   0   0
6   0 0 0 0  90  1356 3670 5700 6165 4945 2997 1365 455 105 15

David Pasino, Table of n, a(n) for n = 1..491

Cf. A059443, A060052.

T(n, k) = Sum{s=0..min(floor(n/2), k)} binomial(n, 2*s) * ((2*s)! / (2^s * s!)) * (-1)^s * A276639(n - 2*s, k - s). (This is the inverse relationship of A276639 in terms of T. A276639(n, k) counts graphs with no isolated points, n points, k edges. The summation range of s, the role of s in the arguments (n - 2s, k - s) of the T or A function being summed, and the coefficient function of s, are the same in the relationship going either way, except that the factor (-1)^s is absent when the function being summed is this T. The coefficient, without the -1, is the number of ways to choose 2s points among the n and group them into s pairs to be s isolated edges. A graph with no isolated points is a graph with some number s of isolated edges and a graph on the complement of the union of those with no isolated edges and no isolated points. That the inverse relationship is almost the same was found empirically for small values of n (leaving k as k), and once found, was readily proved.)

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.

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

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

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A060051 Number of n-block r-bicoverings.

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

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A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

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A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

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