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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.
2
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, 0, 0, 45, 2805, 11550, 3507, 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016, 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884, 0, 0, 0, 0, 0, 1335, 222516, 4710300, 22467312, 24556140, 3026655
OFFSET
1,10
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
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!.
EXAMPLE
Triangle begins:
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, 0, 0, 45, 2805, 11550, 3507;
0, 0, 0, 0, 10, 3595, 59990, 140420, 30016;
0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884;
...
The T(3,3) = 1 matrix is:
[0 1 1]
[1 0 1]
[1 1 0]
The T(4,4) = 3 matrices are:
[0 0 1 1] [0 1 0 1] [0 1 1 0]
[0 0 1 1] [1 0 1 0] [1 0 0 1]
[1 1 0 0] [0 1 0 1] [1 0 0 1]
[1 1 0 0] [1 0 1 0] [0 1 1 0]
PROG
(PARI)
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!)}
T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}
CROSSREFS
Row sums are A370059.
Column sums are A100743.
Main diagonal is A001205.
Cf. A369928, A369932 (unlabeled).
Sequence in context: A161837 A299163 A326128 * A160086 A115869 A115859
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 08 2024
STATUS
approved