0,2

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.

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

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

=====

(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) ) ) } [From Max Alekseyev]

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

Sequence in context: A113664 A077745 A138214 this_sequence A096332 A084284 A084285

Adjacent sequences: A138559 A138560 A138561 this_sequence A138563 A138564 A138565

nonn

Nadia Heninger (nadiah(AT)cs.princeton.edu) and N. J. A. Sloane (njas(AT)research.att.com), May 10 2008

Edited and extended by Max Alekseyev (maxale(AT)gmail.com), May 10 2009