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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. 2

%I

%S 0,1,3,3,6,15,16,12,10,45,110,195,210,120,20,15,105,435,1320,2841,

%T 4410,4845,3360,1350,300,30,21,210,1295,5880,19887,51954,106785,

%U 171360,208565,186375,120855,56805,19110,4410,630,42

%N 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 The length of the rows are 1,1,2,4,7,11,16,22,...: (1+(n-1)*(n-2)/2) = A152947(n).

%C T(n,k) = 0 if k > (n-1)*(n-2)/2 + 1.

%C 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 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 C. W. 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 Triangle starts:

%e ..0

%e ..1

%e ..3......3

%e ..6.....15.....16.....12

%e .10.....45....110....195....210....120.....20

%e .15....105....435...1320...2841...4410...4845...3360...1350....300.....30

%e ...

%Y Sum of n-th row is A245797(n).

%K nonn,tabf

%O 1,3

%A _Chai Wah Wu_, Aug 01 2014

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Last modified June 14 00:02 EDT 2021. Contains 345016 sequences. (Running on oeis4.)