PS on Re: Extend Sequence A005966

[N. J. A. Sloane]

After looking more closely at the source, I see
that Balaban gives a triangle, decomposing the total
numbers of those multigraphs according to the number
of "double bonds".  I'm not quite sure what he means
by this, but here is the triangle:

%I A101204
%S A101204 1,0,1,1,0,1,1,1,2,1,3,4,5,4,1,9,16,22,16,7,1,32,75,112,86,41,10,1,133
%N A101204 Triangle read by rows: T(n,k) = number of planar trivalent (or cubic) multigraphs with 2n nodes and exactly k double bonds, for 0 <= k <= n.
%C A101204 The entries in the first two rows are "by convention".
%D A101204 A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
%O A101204 0,9
%e A101204 Triangle begins
%e A101204 1
%e A101204 0 1
%e A101204 1 0 1
%e A101204 1 1 2 1
%e A101204 3 4 5 4 1
%e A101204 9 16 22 16 7 1
%e A101204 32 75 112 86 41 10 1
%e A101204 133 ...
%K A101204 nonn,tabl
%Y A101204 Row sums give A005966. First column is A005964 (trivalent connected planar graphs with 2n nodes). Second and third columns give A101205, A101206.
%A A101204 njas, Dec 13 2004

A005966 is the sequence Paul Hanna is interested in.  A005964
was already extended by Brendan:

%I A005964 M2816
%S A005964 0,1,1,3,9,32,133,681,3893,24809,169206,1214462,9034509
%N A005964 Number of trivalent connected planar graphs with 2n nodes.
%D A005964 A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976, p. 92.
%H A005964 B. D. McKay, <a href="http://cs.anu.edu.au/~bdm/plantri">Plantri</a>
%H A005964 M. Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a>
%Y A005964 Cf. A058378, A000109.
%K A005964 nonn,nice,hard
%O A005964 1,4
%A A005964 njas
%E A005964 Extended by Brendan McKay (bdm(AT)cs.anu.edu.au) and Gunnar Brinkmann (gunnar(AT)Mathematik.Uni-Bielefeld.DE) using their program "plantri", Dec 19, 2000


The next two columns are new:

%I A101205
%S A101205 1,0,1,4,16,75
%N A101205 Second column of triangle in A0101204.
%D A101205 A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
%K A101205 nonn
%A A101205 njas, Dec 13 2004

%I A101206
%S A101206 1,2,5,22,112
%N A101206 Third column of triangle in A0101204.
%D A101206 A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
%K A101206 nonn
%A A101206 njas, Dec 13 2004

Brendan, can you extend all of these using Plantri ?

NJAS
.