A350746 - OEIS

%I #8 Feb 22 2022 00:33:17

%S 2,3,4,16,18,8,133,155,72,16,1521,1810,910,240,32,22184,26797,14145,

%T 4180,720,64,393681,480879,262514,83230,16520,2016,128,8233803,

%U 10144283,5675866,1888873,409360,58912,5376,256

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

%C The family of quasi-loop-threshold graphs is the smallest family of looped graphs that contains K_1 (a single vertex) and K^loop_1 (a single looped vertex), and is closed under taking unions and adding looped dominating vertices (looped, and adjacent to everything previously added).

%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 See Section 1.4 of Galvin, Wesley and Zacovic link for two methods to compute T(n,k).

%e Triangle begins:

%e         2;

%e         3,        4;

%e        16,       18,       8;

%e       133,      155,      72,      16;

%e      1521,     1810,     910,     240,     32;

%e     22184,    26797,   14145,    4180,    720,    64;

%e    393681,   480879,  262514,   83230,  16520,  2016,  128;

%e   8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256;

%e   ...

%t qltconn[0] = 0; qltconn[1] = 2; qltconn[n_] := qltconn[n] = Sum[StirlingS2[n, k]*(k^(k - 1)), {k, 1, n}] (*qltconn is the number of connected quasi loop threshold graphs on n vertices*); T[n_, l_] := T[n, l] := (Factorial[n]/Factorial[l])*Coefficient[(Sum[(qltconn[k]*(x^k))/Factorial[k], {k, 1, n}])^l, x, n]; Table[T[n, l], {n, 1, 12}, {l, 1, n}]

%Y Row sums are A038052.

%Y Except at n=1, the first column is A048802 (A048802 takes value 1 at n=1).

%K nonn,tabl

%O 1,1

%A _David Galvin_, Jan 13 2022