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A307316 Number of unlabeled leafless loopless multigraphs with n edges.

%I A307316 #30 Feb 09 2024 12:40:18
%S A307316 1,0,1,2,5,11,34,87,279,897,3129,11458,44576,181071,770237,3407332,
%T A307316 15641159,74270464,364014060,1837689540,9540175803,50853577811,
%U A307316 277976050975,1556372791835,8916484189284,52220798342832,312389223102731,1907282708797831,11876576923779692,75376983176576501,487295169002095058
%N A307316 Number of unlabeled leafless loopless multigraphs with n edges.
%C A307316 Multigraphs with no loops and no vertices of degree 1.
%C A307316 The initial terms were computed with Nauty.
%C A307316 Conjecturally, the asymptotic number of completely symmetric polynomials of degree n up to momentum conservation in the limit as the number of particles increases.
%H A307316 Andrew Howroyd, <a href="/A307316/b307316.txt">Table of n, a(n) for n = 0..50</a>
%H A307316 P. T. Komiske, E. M. Metodiev, and J. Thaler, <a href="https://arxiv.org/abs/1911.04491">Cutting Multiparticle Correlators Down to Size</a>, arXiv:1911.04491 [hep-ph], 2019-2020.
%H A307316 Brendan McKay and Adolfo Piperno, <a href="http://pallini.di.uniroma1.it/">nauty and Traces</a>.
%F A307316 Euler transform of A307317.
%e A307316 For n=4 the multigraphs (as sets of edges) are {(0,1),(1,2),(2,3),(3,0)}, {(0,1),(0,1),(1,2),(2,0)}, {(0,1),(0,1),(0,1),(0,1)}, {(0,1),(0,1),(1,2),(1,2)}, and {(0,1),(0,1),(2,3),(2,3)}.
%o A307316 (PARI) \\ See also A370063 for a more efficient program.
%o A307316 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 A307316 edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
%o A307316 seq(n)={my(s=0); forpart(p=2*n, s+=permcount(p)*prod(i=1, #p, 1-x^p[i])/edges(p, w->1-x^w + O(x*x^n))); Vec(s/(2*n)!)} \\ _Andrew Howroyd_, Feb 01 2024
%Y A307316 Conjecturally the same as A226919. Possibly also A254342.
%Y A307316 Row sums of A370063.
%Y A307316 Cf. A050535, A307317 (connected), A369286, A369290 (simple graphs), A369927.
%K A307316 nonn
%O A307316 0,4
%A A307316 _Patrick T. Komiske_, Apr 02 2019
%E A307316 a(0)=1 prepended and a(17) onwards from _Andrew Howroyd_, Feb 01 2024