A139621 -id:A139621 - cp's OEIS frontend

A137975 Row sums of A139621, number of connected directed multigraphs with loops and no vertex of degree 0, with n arcs.

1, 2, 8, 32, 167, 928, 5924, 40211, 293370, 2255406, 18201706, 153176115, 1339271815, 12124484941, 113362749476, 1092329626380, 10827837622018, 110249198676581, 1151562885666429, 12324860339781102, 135026515460855978, 1512882677086123938, 17321462912397361409, 202503301170606347695, 2415733704608822524946, 29387239261415606708127

Offset: 0

Benoit Jubin, May 01 2008, May 10 2008

nonn

Inverse Euler transform of A052171.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A139621.

Cf. A052171, A139627.

\\ See A139621 for G, InvEulerMT.
seq(n)={vecsum([Vec(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])} \\ Andrew Howroyd, Oct 22 2019

Data corrected to match A052171. - R. J. Mathar, Jul 25 2017

A139622 Triangle read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops, with n arcs and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 10, 19, 6, 1, 1, 19, 73, 59, 9, 1, 1, 28, 208, 350, 138, 12, 1, 1, 44, 534, 1670, 1361, 301, 16, 1, 1, 60, 1215, 6476, 9724, 4364, 575, 20, 1, 1, 85, 2542, 21898, 55707, 45284, 12131, 1042, 25, 1, 1, 110, 4951, 65789, 268329, 365063, 175416, 30090, 1749, 30, 1

Offset: 1

Benoit Jubin, May 01 2008

			Triangle begins:
    1
    1    1
    1    2    1
    1    6    4    1
    1   10   19    6    1
    1   19   73   59    9    1
    1   28  208  350  138   12    1
    1   44  534 1670 1361  301   16  1
    ...
T(4 edges, 2 vertices)=6: one graph 1->1, 1->1, 2->1, 1->2; one graph 1->1, 2->1, 2->1, 1->2; one graph 1->1, 1->2, 1->2, 2->1; one graph 1->1, 1->2, 2->1, 2->2; one graph 2->1, 2->1, 2->1, 1->2; one graph 1->2, 1->2, 2->1, 2->1.
T(4 edges, 3 vertices)=4: one graph 1->1, 2->1, 3->2, 1->3; one graph 2->1, 2->1, 3->2, 1->3; one graph 2->1, 3->1, 1->2, 1->3; one graph 2->1, 3->1, 1->2, 2->3.

Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 73.

Row sums are A139627.

Cf. A136564, A139621, A143841, A350489, A350753.

\\ See PARI link in A350489 for program code.
{ my(A=A139622rows(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

T(n,1) = T(n,n) = 1.

T(n,2) = A139621(n,2) - n(n+1)/2.

More terms from R. J. Mathar, Aug 11 2017

Terms a(34) and beyond from Andrew Howroyd, Jan 14 2022

A139625 Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 6, 1, 10, 1, 19, 1, 28, 1, 1, 44, 2, 1, 60, 10, 1, 85, 31, 1, 110, 90, 1, 146, 222, 1, 182, 520, 1, 231, 1090, 1, 1, 280, 2180, 2, 1, 344, 4090, 11, 1

Offset: 1

Benoit Jubin, May 01 2008, Sep 01 2008

Length of the n^th row: floor(sqrt(n)).
These graphs are reflexive (each vertex has a self-loop), so T(n,k) = sum(A139621(n-k^2,m),m=0..k)
T(n,1) = 1, T(n,2) = A005993(n-4), T(n,3) = A050927(n-9), T(n,4) = A050929(n-16).
Row sums: A139630.

			Triangle begins:
  1
  1
  1
  1  1
  1  2
  1  6
  1 10
  1 19
  1 28  1

Cf. A139623, A139624.

A129620 Square array read by falling antidiagonals: T(n,k) is the number of connected directed multigraphs with loops with n arcs and at most k vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 5, 1, 0, 1, 2, 8, 9, 1, 0, 1, 2, 8, 24, 17, 1, 0, 1, 2, 8, 32, 74, 26, 1, 0, 1, 2, 8, 32, 140, 189, 41, 1, 0, 1, 2, 8, 32, 167, 542, 460, 57, 1, 0, 1, 2, 8, 32, 167, 837, 1964, 989, 81, 1, 0, 1, 2, 8, 32, 167, 928, 4167, 6291, 2021, 106, 1, 0

Offset: 0

Benoit Jubin, May 06 2008

Partial sums of the rows of A139621, i.e., T(n,k) = sum(A139621(n,p),p=0..k).

			1  1  1  1  1  1  ...
0  1  2  2  2  2  ...
0  1  5  8  8  8  ...
0  1  9 24 32 32  ...
0  1  17  (...)
(...)

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Cf. A138107, A139621.

T(n,2) = A138107(n,2) - floor(n/2).

If k >= n+1, T(n,k) = A137975(n).

Name edited by M. F. Hasler, Jul 31 2017

Terms a(32) and beyond from Andrew Howroyd, Oct 22 2019

cp's OEIS Frontend

A137975 Row sums of A139621, number of connected directed multigraphs with loops and no vertex of degree 0, with n arcs.

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A139622 Triangle read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops, with n arcs and k vertices.

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A139625 Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

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A129620 Square array read by falling antidiagonals: T(n,k) is the number of connected directed multigraphs with loops with n arcs and at most k vertices.

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