A001374 Number of relational systems on n nodes. Also number of directed 3-multigraphs with loops on n nodes. (Formerly M3717 N1519)

4, 136, 44224, 179228736, 9383939974144, 6558936236286040064, 62879572771326489528942592, 8439543710699844562674685252214784, 16110027001555070629022725866559372785352704, 442829046878106126159584032189649757399796014050181120

Offset

1,1

References

W. Oberschelp, "Strukturzahlen in endlichen Relationssystemen", in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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 = 1..40

W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]

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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*4^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Array[a, 15] (* Jean-François Alcover, Jul 08 2018, 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, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*4^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017
(Python)
from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A001374(n): return int(sum(Fraction(1<<((sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))<<1)+sum(q*r**2 for q, r in p.items())<<1), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 10 2024

Crossrefs

Cf. A000595, A004105, A053468, A053516.

Sequence in context: A054052 A012070 A366446 * A229416 A155207 A201388

Adjacent sequences: A001371 A001372 A001373 * A001375 A001376 A001377

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 14 2000

Status

approved