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 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 Andrew Howroyd, Table of n, a(n) for n = 0..50 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 Adjacent sequences: A307313 A307314 A307315 * A307317 A307318 A307319 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

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