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A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

%I A369931 #11 Oct 15 2024 00:01:54
%S A369931 0,0,0,0,0,1,0,0,0,3,0,0,0,6,12,0,0,0,1,85,70,0,0,0,0,100,990,465,0,0,
%T A369931 0,0,45,2805,11550,3507,0,0,0,0,10,3595,59990,140420,30016,0,0,0,0,1,
%U A369931 2697,147441,1174670,1802682,286884,0,0,0,0,0,1335,222516,4710300,22467312,24556140,3026655
%N A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.
%C A369931 T(n,k) is the number of traceless symmetric binary matrices with 2n 1's and k rows and at least two 1's in every row.
%H A369931 Andrew Howroyd, <a href="/A369931/b369931.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A369931 T(n,k) = k!*[x^k][y^n] exp(y*x^2/2 - x) * Sum_{j>=0} (1 + y)^binomial(j, 2)*(x/exp(y*x))^j/j!.
%e A369931 Triangle begins:
%e A369931   0;
%e A369931   0, 0;
%e A369931   0, 0, 1;
%e A369931   0, 0, 0, 3;
%e A369931   0, 0, 0, 6,  12;
%e A369931   0, 0, 0, 1,  85,   70;
%e A369931   0, 0, 0, 0, 100,  990,    465;
%e A369931   0, 0, 0, 0,  45, 2805,  11550,    3507;
%e A369931   0, 0, 0, 0,  10, 3595,  59990,  140420,   30016;
%e A369931   0, 0, 0, 0,   1, 2697, 147441, 1174670, 1802682, 286884;
%e A369931   ...
%e A369931 The T(3,3) = 1 matrix is:
%e A369931   [0 1 1]
%e A369931   [1 0 1]
%e A369931   [1 1 0]
%e A369931 The T(4,4) = 3 matrices are:
%e A369931   [0 0 1 1]  [0 1 0 1]  [0 1 1 0]
%e A369931   [0 0 1 1]  [1 0 1 0]  [1 0 0 1]
%e A369931   [1 1 0 0]  [0 1 0 1]  [1 0 0 1]
%e A369931   [1 1 0 0]  [1 0 1 0]  [0 1 1 0]
%o A369931 (PARI)
%o A369931 G(n)={my(A=x/exp(x*y + O(x*x^n))); exp(y*x^2/2 - x + O(x*x^n)) * sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*A^k/k!)}
%o A369931 T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}
%Y A369931 Row sums are A370059.
%Y A369931 Column sums are A100743.
%Y A369931 Main diagonal is A001205.
%Y A369931 Cf. A369928, A369932 (unlabeled).
%K A369931 nonn,tabl
%O A369931 1,10
%A A369931 _Andrew Howroyd_, Feb 08 2024