OFFSET
0,9
COMMENTS
Here, a k-multidigraph is a directed graph where up to k arcs (directed edges) are allowed to join vertex pairs. The arcs have no identity, i.e., they are indistinguishable except for the ordered pair of distinct vertices that they join.
LINKS
Seiichi Manyama, Antidiagonals n = 0..50, flattened
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178.
FORMULA
Let E(x) = Sum_{n>=0} x^n/(n!*(k+1)^binomial(n,2)). Then 1/E(-x) = Sum_{n>=0} T(n,k)x^n/(n!*(k+1)^binomial(n,2)).
T(0,k) = 1 and T(n,k) = Sum_{j=1..n} (-1)^(j+1) * (k+1)^(j*(n-j)) * binomial(n,j) * T(n-j,k) for n > 0. - Seiichi Manyama, Jun 13 2022
EXAMPLE
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 25, 109, 289, 601, 1081, ...
1, 543, 9449, 63487, 267249, 849311, ...
1, 29281, 3068281, 69711361, 742650001, 5004309601, ...
MATHEMATICA
nn = 5; Table[g[n_] := q^Binomial[n, 2] n!; e[z_] := Sum[z^k/g[k], {k, 0, nn}];
Table[g[n], {n, 0, nn}] CoefficientList[Series[1/e[-z], {z, 0, nn}], z], {q, 1, nn + 1}] //Transpose // Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 21 2021
STATUS
approved