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
[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.
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\\ 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
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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
A060489
Number of 5-block ordered tricoverings of an unlabeled n-set.
Original entry on oeis.org
0, 0, 60, 375, 1392, 4020, 9960, 22200, 45730, 88543, 163000, 287650, 489610, 807625, 1295944, 2029165, 3108220, 4667690, 6884660, 9989345, 14277740, 20126570, 28010840, 38524310, 52403246, 70553825, 94083600, 124337460, 162938550, 211834647, 273350520, 350246835
Offset: 1
a(1)=a(2)=0 prepended and terms a(30) and beyond from
Andrew Howroyd, Jan 30 2020
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
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;
...
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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 *)
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\\ 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
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