A004103 Number of nets on n unlabeled nodes. (Formerly M1942)

1, 2, 9, 56, 705, 19548, 1419237, 278474976, 148192635483, 213558945249402, 836556995284293897, 8962975658381123937708, 264404516190234685662666051, 21610417954162750247842392794292, 4921335335427778307286708119839406529, 3138313838161414849743136458064895837170596

Offset

0,2

Comments

A net in this context is a graph with both signed vertices and signed edges. - Andrew Howroyd, Sep 25 2018

References

F. Harary and R. W. Robinson, Exposition of the enumeration of point-line-signed graphs, pp. 19 - 33 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.

R. W. Robinson, personal communication.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..22 from R. W. Robinson)

Frank Harary, Edgar M. Palmer, Robert W. Robinson, Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.

R. W. Robinson, Notes - "A Present for Neil Sloane"

R. W. Robinson, Notes - computer printout

Mathematica

permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 16, 0] (* Jean-François Alcover, Aug 17 2019, after Andrew Howroyd *)

Prog

(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ Andrew Howroyd, Sep 25 2018

Crossrefs

Cf. A004102 (signed edges only), A000666 (signed vertices only).

Cf. A004107.

Sequence in context: A277482 A274393 A371196 * A295775 A223381 A175896

Adjacent sequences: A004100 A004101 A004102 * A004104 A004105 A004106

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(0)=1 prepended and a(13)-a(14) added by Andrew Howroyd, Sep 25 2018

Status

approved