A320490
Inverse Euler transform of A004106.
Original entry on oeis.org
1, 2, 0, 4, 16, 98, 766, 9982, 205574, 6546960, 352380815, 29668304472, 4303531875334, 985391751078694, 391866509413730125, 248491878411568471280, 274544424465173973822126, 488310442484985365942048784, 1513516279388226700191978571979
Offset: 0
A004107
Number of self-dual nets with 2n nodes.
Original entry on oeis.org
1, 1, 9, 165, 24651, 29522961, 286646256675, 21717897090413481, 12980536689318626076840, 62082697145168772833294318409, 2405195296608025717214293025492960466, 762399078635131851885116768114137369439908725
Offset: 0
- 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).
- Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 1..13 from R. W. Robinson)
- Frank Harary, Edgar M. Palmer, Robert W. Robinson, and 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
-
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_] := 2 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 Quotient[v[[i]], 2], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 12, 0] (* Jean-François Alcover, Aug 17 2019, after Andrew Howroyd *)
-
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) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2*2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 25 2018
-
from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A004107(n): return int(sum(Fraction(3**((sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))<<1)+sum(((q&-2)+q*(r-1))*r for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024
A320491
Number of connected edge-self-dual nets with n nodes.
Original entry on oeis.org
1, 2, 0, 4, 13, 98, 748, 9982, 205290, 6546960, 352371887, 29668304472, 4303531187466, 985391751078694, 391866509275972993, 248491878411568471280, 274544424465100159954078, 488310442484985365942048784, 1513516279388226593561422201905
Offset: 0
Dead sequence restored, corrected and extended by
Andrew Howroyd, Jan 28 2020
Showing 1-3 of 3 results.
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