cp's OEIS Frontend

A350531 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.

%I A350531 #19 Feb 04 2022 09:55:15
%S A350531 2,3,3,13,9,4,75,52,18,5,541,375,130,30,6,4683,3246,1125,260,45,7,
%T A350531 47293,32781,11361,2625,455,63,8,545835,378344,131124,30296,5250,728,
%U A350531 84,9,7087261,4912515,1702548,393372,68166,9450,1092,108,10
%N A350531 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.
%C A350531 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 A350531 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 A350531 T(n,n) = n+1 for n >= 1; T(n,1) = Sum_{j=0..n-1} A173018(n,j)*2^j for n >= 2; T(n,k) = binomial(n, k-1)*T(n-k+1,1) for n >= 3, 2 <= k <= n-1.
%e A350531 Triangle begins:
%e A350531         2;
%e A350531         3,       3;
%e A350531        13,       9,       4;
%e A350531        75,      52,      18,      5;
%e A350531       541,     375,     130,     30,     6;
%e A350531      4683,    3246,    1125,    260,    45,    7;
%e A350531     47293,   32781,   11361,   2625,   455,   63,    8;
%e A350531    545835,  378344,  131124,  30296,  5250,  728,   84,   9;
%e A350531   7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 10;
%e A350531   ...
%t A350531 eulerian[n_, m_] := eulerian[n, m] =
%t A350531   Sum[((-1)^k)*Binomial[n+1, k]*((m+1-k)^n), {k, 0, m+1}] (* eulerian[n, m] is an Eulerian number, counting the permutations of [n] with m descents *); T[1, 1] = 2; T[n_, 1] := T[n, 1] = Sum[eulerian[n, k]*(2^k), {k, 0, n - 1}]; T[n_, n_] := T[n, n] = n + 1; T[n_, k_] := T[n, k] = Binomial[n, k - 1]*T[n - k + 1, 1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]
%Y A350531 Row sums are A000629.
%Y A350531 Except at n = 1, first column is A000670.
%Y A350531 Essentially the same as A154921 --- in A350531 (this triangle), replace the last nonzero entry in row m (this entry is m+1) with the two entries m, 1 to get A154921.
%Y A350531 Cf. A173018.
%K A350531 nonn,tabl
%O A350531 1,1
%A A350531 _David Galvin_, Jan 03 2022