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A369928 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled vertices with k edges and without endpoints, n >= 0, 0 <= k <= n*(n-1)/2.

%I A369928 #11 Feb 07 2024 20:26:31
%S A369928 1,1,1,0,1,0,0,1,1,0,0,4,3,6,1,1,0,0,10,15,42,90,100,45,10,1,1,0,0,20,
%T A369928 45,162,595,1590,3075,3655,2703,1335,455,105,15,1,1,0,0,35,105,462,
%U A369928 2310,9495,32130,85365,166341,231861,237125,184380,111870,53634,20307,5985,1330,210,21,1
%N A369928 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled vertices with k edges and without endpoints, n >= 0, 0 <= k <= n*(n-1)/2.
%H A369928 Andrew Howroyd, <a href="/A369928/b369928.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)
%F A369928 T(n,k) = A084546(n,k) - A245796(n,k).
%F A369928 E.g.f.: exp(y*x^2/2) * Sum_{k>=0} (1 + y)^binomial(k, 2)*(x/exp(y*x))^k/k!.
%e A369928 Triangle begins:
%e A369928 [0] 1;
%e A369928 [1] 1;
%e A369928 [2] 1, 0;
%e A369928 [3] 1, 0, 0,  1;
%e A369928 [4] 1, 0, 0,  4,  3,  6,    1;
%e A369928 [5] 1, 0, 0, 10, 15,  42,  90,  100,   45,   10,    1;
%e A369928 [6] 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1;
%o A369928 (PARI) \\ row(n) gives n-th row as vector.
%o A369928 row(n)={my(A=x/exp(x*y + O(x*x^n))); Vecrev(polcoef(serlaplace(exp(y*x^2/2 + O(x*x^n)) * sum(k=0, n, (1 + y)^binomial(k, 2)*A^k/k!)), n), 1 + binomial(n,2))}
%o A369928 { for(n=0, 6, print(row(n))) }
%Y A369928 Row sums are A059167.
%Y A369928 Cf. A084546, A123551 (unlabeled), A245796 (with endpoints).
%K A369928 nonn,tabf
%O A369928 0,12
%A A369928 _Andrew Howroyd_, Feb 07 2024