A350528 - OEIS

%I #11 Jan 12 2022 21:40:21

%S 1,1,1,4,3,1,23,19,6,1,181,155,55,10,1,1812,1591,600,125,15,1,22037,

%T 19705,7756,1750,245,21,1,315569,286091,116214,27741,4270,434,28,1,

%U 5201602,4766823,1983745,493794,81291,9198,714,36,1

%N Triangle read by rows: T(n,k) is the number of labeled quasi-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.

%C The family of quasi-threshold graphs is the smallest family of graphs that contains K_1 (a single vertex), and is closed under taking unions and adding dominating vertices (adjacent to all other vertices).

%H D. Galvin, G. Wesley and B. Zacovic, <a href="https://arxiv.org/abs/2110.08953">Enumerating threshold graphs and some related graph classes</a>, arXiv:2110.08953 [math.CO], 2021.

%F T(n,k) = Sum_{j=1..n} (-1)^(n-j)*Stirling2(n, j)*k*binomial(j, k)*j^(j-k-1) for n >= 1, 1 <= k <= n.

%e Triangle begins:

%e        1;

%e        1,      1;

%e        4,      3,      1;

%e       23,     19,      6,      1;

%e      181,    155,     55,     10,     1;

%e     1812,   1591,    600,    125,    15,    1;

%e    22037,  19705,   7756,   1750,   245,   21,   1;

%e   315569, 286091; 116214,  27741,  4270,  434,  28,  1;

%e   ...

%t T[n_, k_] := T[n, k] = Sum[((-1)^(n - j))*StirlingS2[n, j]*k*Binomial[j, k]*(j^(j - k - 1)), {j, 1, n}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]

%Y First column is A058863.

%Y Row sums are A058864.

%Y Cf. A008277.

%K nonn,tabl

%O 1,4

%A _David Galvin_, Jan 03 2022