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A350745 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k loops, for n >= 0 and 0 <= k <= n.

%I A350745 #22 May 06 2023 11:13:01
%S A350745 1,1,1,1,4,1,1,12,12,1,1,32,84,32,1,1,80,460,460,80,1,1,192,2190,4600,
%T A350745 2190,192,1,1,448,9534,37310,37310,9534,448,1,1,1024,39032,264208,
%U A350745 483140,264208,39032,1024,1,1,2304,152856,1702344,5229756,5229756,1702344,152856,2304,1
%N A350745 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k loops, for n >= 0 and 0 <= k <= n.
%C A350745 Loop-threshold graphs are constructed from either a single unlooped vertex or a single looped vertex by iteratively adding isolated vertices (adjacent to nothing previously added) and looped dominating vertices (looped, and adjacent to everything previously added).
%H A350745 Andrew Howroyd, <a href="/A350745/b350745.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50).
%H A350745 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 A350745 T(n,0) = 1; T(n,k) = binomial(n,k) * Sum_{j=1..n} j!*Stirling2(k,j) * ((j-1)! * Stirling2(n-k,j-1) + 2*j!*Stirling2(n-k,j) + (j+1)!*Stirling2(n-k,j+1)).
%F A350745 T(n,k) = T(n,n-k).
%F A350745 Sum_{k=0..2*n} (-1)^k * T(2*n,k) = A210657(n). - _Alois P. Heinz_, Feb 01 2022
%e A350745 Triangle begins:
%e A350745   1;
%e A350745   1,    1;
%e A350745   1,    4,      1;
%e A350745   1,   12,     12,       1;
%e A350745   1,   32,     84,      32,       1;
%e A350745   1,   80,    460,     460,      80,       1;
%e A350745   1,  192,   2190,    4600,    2190,     192,       1;
%e A350745   1,  448,   9534,   37310,   37310,    9534,     448,      1;
%e A350745   1, 1024,  39032,  264208,  483140,  264208,   39032,   1024,    1;
%e A350745   1, 2304, 152856, 1702344, 5229756, 5229756, 1702344, 152856, 2304, 1;
%e A350745   ...
%t A350745 T[n_, 0] := T[n, 0] = 1; T[n_, k_] := T[n, k] = Binomial[n, k]*Sum[Factorial[l]*StirlingS2[k, l]*(Factorial[l - 1]*StirlingS2[n - k, l - 1] + 2*Factorial[l]*StirlingS2[n - k, l] + Factorial[l + 1]*StirlingS2[n - k, l + 1]), {l, 1, n + 1}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]
%o A350745 (PARI) T(n,k) = if(k==0, 1, binomial(n,k) * sum(j=1, n, j!*stirling(k,j,2) * ((j-1)! * stirling(n-k,j-1,2) + 2*j!*stirling(n-k,j,2) + (j+1)!*stirling(n-k,j+1,2)))) \\ _Andrew Howroyd_, May 06 2023
%Y A350745 Row sums are A000629.
%Y A350745 Columns k=0..1 give: A000012, A001787,
%Y A350745 Cf. A210657.
%K A350745 nonn,tabl
%O A350745 0,5
%A A350745 _David Galvin_, Jan 13 2022