The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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
P. T. Komiske, E. M. Metodiev, and J. Thaler, Cutting Multiparticle Correlators Down to Size, arXiv:1911.04491 [hep-ph], 2019-2020.
Brendan McKay and Adolfo Piperno, nauty and Traces.
FORMULA
Euler transform of A307317.
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[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
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)))}
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
CROSSREFS
Conjecturally the same as A226919. Possibly also A254342.
Row sums of A370063.
Cf. A050535, A307317 (connected), A369286, A369290 (simple graphs), A369927.
Sequence in context: A254342 A080068 A226919 * A298122 A196690 A101834
KEYWORD
nonn
AUTHOR
Patrick T. Komiske, Apr 02 2019
EXTENSIONS
a(0)=1 prepended and a(17) onwards from Andrew Howroyd, Feb 01 2024
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 18:19 EDT 2024. Contains 372840 sequences. (Running on oeis4.)