

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



0, 1, 3, 3, 6, 15, 16, 12, 10, 45, 110, 195, 210, 120, 20, 15, 105, 435, 1320, 2841, 4410, 4845, 3360, 1350, 300, 30, 21, 210, 1295, 5880, 19887, 51954, 106785, 171360, 208565, 186375, 120855, 56805, 19110, 4410, 630, 42
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OFFSET

1,3


COMMENTS

The length of the rows are 1,1,2,4,7,11,16,22,...: (1+(n1)*(n2)/2) = A152947(n).
T(n,k) = 0 if k > (n1)*(n2)/2 + 1.
Let j = (n1)*(n2)/2. For i >=0, n >= 4+i, T(n,ji+1) = n*(n1)*binomial(j,i).
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(n1), T(n,2) = A050534(n) and T(n,3) = A053526(n).


LINKS

Table of n, a(n) for n=1..42.
C. W. Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quantph], 2014.


EXAMPLE

Triangle starts:
..0
..1
..3......3
..6.....15.....16.....12
.10.....45....110....195....210....120.....20
.15....105....435...1320...2841...4410...4845...3360...1350....300.....30
...


CROSSREFS

Sum of nth row is A245797(n).
Sequence in context: A143418 A336452 A092370 * A006807 A298180 A119460
Adjacent sequences: A245793 A245794 A245795 * A245797 A245798 A245799


KEYWORD

nonn,tabf


AUTHOR

Chai Wah Wu, Aug 01 2014


STATUS

approved



