A321256 -id:A321256 - cp's OEIS frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

cp's OEIS Frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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