

A307316


Number of unlabeled leafless loopless multigraphs with n edges.


6



1, 0, 1, 2, 5, 11, 34, 87, 279, 897, 3129, 11458, 44576, 181071, 770237, 3407332, 15641159, 74270464, 364014060, 1837689540, 9540175803, 50853577811, 277976050975, 1556372791835, 8916484189284, 52220798342832, 312389223102731, 1907282708797831, 11876576923779692, 75376983176576501, 487295169002095058
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OFFSET

0,4


COMMENTS

Multigraphs with no loops and no vertices of degree 1.
The initial terms were computed with Nauty.
Conjecturally, the asymptotic number of completely symmetric polynomials of degree n up to momentum conservation in the limit as the number of particles increases.


LINKS



FORMULA



EXAMPLE

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)}.


PROG

(PARI) \\ See also A370063 for a more efficient program.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c1)\2)*if(c%2, 1, t(c/2)))}
seq(n)={my(s=0); forpart(p=2*n, s+=permcount(p)*prod(i=1, #p, 1x^p[i])/edges(p, w>1x^w + O(x*x^n))); Vec(s/(2*n)!)} \\ Andrew Howroyd, Feb 01 2024


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



