A053513 - OEIS

%I #17 Jul 09 2024 19:11:06

%S 3,18,165,3132,137268,15548004,4679446950,3771927027864,

%T 8186669639820081,48184182482857319682,774912347548961791914585,

%U 34299111628183837790980740312,4205499936656520106909422649497294

%N Number of 2-multigraphs with loops on n nodes.

%C A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).

%H Andrew Howroyd, <a href="/A053513/b053513.txt">Table of n, a(n) for n = 1..50</a>

%t 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];

%t edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2] + 1];

%t a[n_] := (s=0; Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!);

%t Array[a, 15] (* _Jean-François Alcover_, Jul 08 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o 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}

%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2 + 1)}

%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 22 2017

%o (Python)

%o from itertools import combinations

%o from math import prod, gcd, factorial

%o from fractions import Fraction

%o from sympy.utilities.iterables import partitions

%o def A053513(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum(((q>>1)+1)*r+(q*r*(r-1)>>1) 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

%Y Cf. A000666, A004102, A053514.

%K easy,nonn

%O 1,1

%A _Vladeta Jovovic_, Jan 14 2000