cp's OEIS Frontend

A245796 T(n,k) is the number of labeled graphs of n vertices and k edges that have endpoints, where an endpoint is a vertex with degree 1.

%I A245796 #23 Jul 24 2023 04:52:41
%S A245796 0,1,3,3,6,15,16,12,10,45,110,195,210,120,20,15,105,435,1320,2841,
%T A245796 4410,4845,3360,1350,300,30,21,210,1295,5880,19887,51954,106785,
%U A245796 171360,208565,186375,120855,56805,19110,4410,630,42
%N A245796 T(n,k) is the number of labeled graphs of n vertices and k edges that have endpoints, where an endpoint is a vertex with degree 1.
%C A245796 The length of the rows are 1,1,2,4,7,11,16,22,...: (1+(n-1)*(n-2)/2) = A152947(n).
%C A245796 T(n,k) = 0 if k > (n-1)*(n-2)/2 + 1.
%C A245796 Let j = (n-1)*(n-2)/2.  For i >=0, n >= 4+i, T(n,j-i+1) = n*(n-1)*binomial(j,i).
%C A245796 For k <= 3, T(n,k) is equal to the number of labeled bipartite graphs with n vertices and k edges.  In particular, T(n,1) = A000217(n-1), T(n,2) = A050534(n) and T(n,3) = A053526(n).
%H A245796 Chai Wah Wu, <a href="http://arxiv.org/abs/1407.5663">Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics</a>, arXiv:1407.5663 [quant-ph], 2014.
%e A245796 Triangle starts:
%e A245796 ..0
%e A245796 ..1
%e A245796 ..3......3
%e A245796 ..6.....15.....16.....12
%e A245796 .10.....45....110....195....210....120.....20
%e A245796 .15....105....435...1320...2841...4410...4845...3360...1350....300.....30
%e A245796 ...
%Y A245796 Sum of n-th row is A245797(n).
%Y A245796 Cf. A000217, A050534, A053526, A152947.
%K A245796 nonn,tabf
%O A245796 1,3
%A A245796 _Chai Wah Wu_, Aug 01 2014