A369928 - OEIS

%I #11 Feb 07 2024 20:26:31

%S 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 45,162,595,1590,3075,3655,2703,1335,455,105,15,1,1,0,0,35,105,462,

%U 2310,9495,32130,85365,166341,231861,237125,184380,111870,53634,20307,5985,1330,210,21,1

%N 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 Andrew Howroyd, <a href="/A369928/b369928.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)

%F T(n,k) = A084546(n,k) - A245796(n,k).

%F E.g.f.: exp(y*x^2/2) * Sum_{k>=0} (1 + y)^binomial(k, 2)*(x/exp(y*x))^k/k!.

%e Triangle begins:

%e [0] 1;

%e [1] 1;

%e [2] 1, 0;

%e [3] 1, 0, 0,  1;

%e [4] 1, 0, 0,  4,  3,  6,    1;

%e [5] 1, 0, 0, 10, 15,  42,  90,  100,   45,   10,    1;

%e [6] 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1;

%o (PARI) \\ row(n) gives n-th row as vector.

%o 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 { for(n=0, 6, print(row(n))) }

%Y Row sums are A059167.

%Y Cf. A084546, A123551 (unlabeled), A245796 (with endpoints).

%K nonn,tabf

%O 0,12

%A _Andrew Howroyd_, Feb 07 2024