A060091 -id:A060091 - cp's OEIS frontend

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

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!.

A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

4, 13, 28, 50, 80, 119, 168, 228, 300, 385, 484, 598, 728, 875, 1040, 1224, 1428, 1653, 1900, 2170, 2464, 2783, 3128, 3500, 3900, 4329, 4788, 5278, 5800, 6355, 6944, 7568, 8228, 8925, 9660, 10434, 11248, 12103, 13000, 13940, 14924, 15953, 17028, 18150, 19320

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.
If Y is a 4-subset of an n-set X then, for n>=6, a(n-3) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007
Also the number of balls in a triangular pyramid of which all balls located on the edges have been removed such that the remaining pyramid's edges each consist of two adjacent balls. The layers of pyramids of this form start (from the top) 3, 7, 12, 18, 25, 33,... (A055998) with one smaller additional layer 1, 3, 6, 10, 15, 21,... (A000217) at the bottom. Thus, a(n) = A000217(n) + Sum_{k=1..n} A055998(k). Example: a(4) = (3+7+12+18)+10 = 50. - K. G. Stier, Dec 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Essentially the same as A026054. - Vladeta Jovovic, Jun 15 2006
Column k=4 of A060492.
Cf. A060091, A060491.
Fourth column (m=3) of (1, 4)-Pascal triangle A095666.

[(n-2)*(n-1)*(n+9)/6: n in [3..60]]; // Vincenzo Librandi, Jun 15 2011

Table[ 3 (n - 1) (n - 2)/2! + n (n - 1) (n - 2)/3!, {n, 3, 62}] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
Table[(n-2)(n-1)(n+9)/6,{n,3,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {4,13,28,50},50] (* Harvey P. Dale, Jul 21 2012 *)

a(n)=(n-2)*(n-1)*(n+9)/6 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n+3, 3) - 6*binomial(n+1, 1) + 8*binomial(n, 0) - 3*binomial(n-1, -1).
G.f.: -y^3*(-4+3*y)/(-1+y)^4.
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, 1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k! ).
a(n) = (n+9)*binomial(n-1, 2)/3.
a(n) = (n-2)*(n-1)*(n+9)/6. - Zak Seidov, Jun 15 2006
a(3)=4, a(4)=13, a(5)=28, a(6)=50, a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 21 2012

A290776 Triangle T(n,k) read by rows: the number of connected, loopless, non-oriented, vertex-labeled graphs with n >= 0 edges and k >= 1 vertices, allowing multi-edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 16, 0, 1, 12, 63, 125, 0, 1, 18, 162, 722, 1296, 0, 1, 25, 341, 2565, 10140, 16807, 0, 1, 33, 636, 7180, 47100, 169137, 262144, 0, 1, 42, 1092, 17335, 168285, 987567, 3271576, 4782969, 0, 1, 52, 1764, 37750, 509545, 4364017, 23315936, 72043092, 100000000

Offset: 0

R. J. Mathar, Aug 10 2017

This is the vertex-labeled companion to A191646.

			The triangle starts in row n=0 with 1 <= k <= n+1 vertices as
  1;
  0, 1;
  0, 1,  3;
  0, 1,  7,   16;
  0, 1, 12,   63,   125;
  0, 1, 18,  162,   722,   1296;
  0, 1, 25,  341,  2565,  10140,   16807;
  0, 1, 33,  636,  7180,  47100,  169137,   262144;
  0, 1, 42, 1092, 17355, 168285,  987567,  3271576,  4782969;
  0, 1, 52, 1764, 37750, 509545, 4364017, 23315936, 72043092, 100000000;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 62.

Cf. A055998 (k=3), A000272 (diagonal), A060091 (k=4?), A060093 (k=5?).

S[m_, n_] := Binomial[Binomial[m, 2] + n - 1, n];
R[nn_] := Module[{cc = Array[0&, {nn, nn}]}, cc[[1, 1]] = 1; For[m = 1, m <= nn, m++, For[n = 1, n <= nn-1, n++, cc[[m, n+1]] = S[m, n] - S[m-1, n] - Sum[Sum[Binomial[m-1, i-1]*cc[[i, j+1]]*S[m-i, n-j], {j, 1, n}], {i, 2, m-1}]]]; cc // Transpose];
A = R[10];
Table[A[[n, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 13 2018, after Andrew Howroyd *)

\\ here S(m,n) is m nodes with n edges, not necessarily connected
S(m,n)={ binomial(binomial(m,2) + n - 1, n) }
R(N)={ my(C=matrix(N,N)); C[1,1]=1; for(m=1, N, for(n=1, N-1, C[m,n+1] = S(m,n) - S(m-1,n) - sum(i=2, m-1, sum(j=1, n, binomial(m-1, i-1)*C[i,j+1]*S(m-i, n-j))))); C~; }
{ my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k],", ")); print) } \\ Andrew Howroyd, May 13 2018

Terms a(34) and beyond from Andrew Howroyd, May 13 2018

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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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A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.

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A290776 Triangle T(n,k) read by rows: the number of connected, loopless, non-oriented, vertex-labeled graphs with n >= 0 edges and k >= 1 vertices, allowing multi-edges.

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