A138562 Number of "squashed-tree" graphs with n central nodes, the labeled case, allowing the direct link between L and R.

1, 4, 38, 616, 14744, 479364, 20021768, 1031673164, 63597989864, 4579513525216, 377953469391584, 35211153592004064, 3657198048669038384, 419166387797337858500, 52561549979435515611488, 7158828855330149502246076, 1052478318277669232896492064, 166132533639153074372662711680

Offset: 0

Author

Nadia Heninger and N. J. A. Sloane, May 10 2008

Keywords

• nonn

Comments

These are simple connected graphs with n+2 nodes labeled L, R, 1, 2, ..., n. The subgraph on nodes 1..n is a forest (no loops). Nodes L and R are both connected to some subset of 1..n and perhaps to each other.

These are the graphs that can arise when one starts with a tree with m >= n+2 labeled nodes, some of which are colored blue, some are colored red and the remaining n nodes are uncolored. Then all the blue nodes are coalesced into a single node L and all the red nodes into a single node R.

Examples

			a(0) = 1: L--R.
a(1) = 4: L--1--R, 1--L--R, L--R--1 and the 3-cycle L--1--R--L.
a(2) = 38: the 14 examples shown in A138460 plus the same set with an edge joining L and R: 28 in all, plus the following 10 graphs, for a total of 38.
=====
. 1
./..
L---R (number = 2)
.\..
. 2
=====
. 1
./..
L---R (number = 2)
.../
. 2
=====
. 1
./|.
L-|-R (number = 2)
.\|.
. 2
=====
. 1
./|.
L-|-R (number = 4)
..|.
. 2
=====
		

Crossrefs

Cf. A138560. A001187(n+2) is an upper bound.

Programs

PARI

{ a(n) = local(p,q,m); p=partitions(n); sum(j=1,#p, q=p[j]; m=vector(n); for(i=1,#q,m[q[i]]++); n! * prod(i=1,#q,q[i]^(q[i]-2)/q[i]!) / prod(i=1,#m,m[i]!) * (prod(i=1,#q,4^q[i]-1)*2 - 2^#q*prod(i=1,#q,2^q[i]-1) ) ) } \\ Max Alekseyev, May 10 2009

Formula

Although we have not written out all the details of the proof, it appears that a(n) ~ 2^n*n^(n-2).

Extensions

Edited and extended by Max Alekseyev, May 10 2009