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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.
2

%I #11 Oct 15 2024 00:01:54

%S 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 0,0,45,2805,11550,3507,0,0,0,0,10,3595,59990,140420,30016,0,0,0,0,1,

%U 2697,147441,1174670,1802682,286884,0,0,0,0,0,1335,222516,4710300,22467312,24556140,3026655

%N 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 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 Andrew Howroyd, <a href="/A369931/b369931.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%F 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 Triangle begins:

%e 0;

%e 0, 0;

%e 0, 0, 1;

%e 0, 0, 0, 3;

%e 0, 0, 0, 6, 12;

%e 0, 0, 0, 1, 85, 70;

%e 0, 0, 0, 0, 100, 990, 465;

%e 0, 0, 0, 0, 45, 2805, 11550, 3507;

%e 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016;

%e 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884;

%e ...

%e The T(3,3) = 1 matrix is:

%e [0 1 1]

%e [1 0 1]

%e [1 1 0]

%e The T(4,4) = 3 matrices are:

%e [0 0 1 1] [0 1 0 1] [0 1 1 0]

%e [0 0 1 1] [1 0 1 0] [1 0 0 1]

%e [1 1 0 0] [0 1 0 1] [1 0 0 1]

%e [1 1 0 0] [1 0 1 0] [0 1 1 0]

%o (PARI)

%o 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 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 Row sums are A370059.

%Y Column sums are A100743.

%Y Main diagonal is A001205.

%Y Cf. A369928, A369932 (unlabeled).

%K nonn,tabl

%O 1,10

%A _Andrew Howroyd_, Feb 08 2024