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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

Let j = (n-1)*(n-2)/2.  For i >=0, n >= 4+i, T(n,j-i+1) = n*(n-1)*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(n-1), 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 [quant-ph], 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 n-th 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

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Last modified April 16 23:40 EDT 2021. Contains 343051 sequences. (Running on oeis4.)