The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A020555 Number of multigraphs on n labeled edges (with loops). Also number of genetically distinct states amongst n individuals. 28
 1, 2, 9, 66, 712, 10457, 198091, 4659138, 132315780, 4441561814, 173290498279, 7751828612725, 393110572846777, 22385579339430539, 1419799938299929267, 99593312799819072788, 7678949893962472351181, 647265784993486603555551, 59357523410046023899154274 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also the number of factorizations of (p_n#)^2. - David W. Wilson, Apr 30 2001 Also the number of multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018 a(n) gives the number of genetically distinct states for n diploid individuals in the case that maternal and paternal alleles transmitted to the individuals are not distinguished (if maternal and paternal alleles are distinguished, then the number of states is A000110(2n)). - Noah A Rosenberg, Aug 23 2022 REFERENCES D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018 E. Keith Lloyd, Math. Proc. Camb. Phil. Soc., vol. 103 (1988), 277-284. A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000. G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..310 (first 101 terms from Vincenzo Librandi) G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248. G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission] Marko Riedel et al., Set partitions of {1,1,2,2,...,n,n} E. A. Thompson, Gene identities and multiple relationships. Biometrics 30 (1974), 667-680. See Table 5. FORMULA Lloyd's article gives a complicated explicit formula. E.g.f.: exp(-3/2 + exp(x)/2)*Sum_{n>=0} exp(binomial(n+1, 2)*x)/n! [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004 a(n) = A001055(A002110(n)^2). - Alois P. Heinz, Aug 23 2022 EXAMPLE From Gus Wiseman, Jul 18 2018: (Start) The a(2) = 9 multiset partitions of {1, 1, 2, 2}: (1122), (1)(122), (2)(112), (11)(22), (12)(12), (1)(1)(22), (1)(2)(12), (2)(2)(11), (1)(1)(2)(2). (End) MAPLE B := n -> combinat[bell](n): P := proc(m, n) local k; global B; option remember; if n = 0 then B(m) else (1/2)*( P(m+2, n-1) + P(m+1, n-1) + add( binomial(n-1, k)*P(m, k), k=0..n-1) ); fi; end; r:=m->[seq(P(m, n), n=0..20)]; r(0); # N. J. A. Sloane, Dec 30 2018 MATHEMATICA max = 16; s = Series[Exp[-3/2 + Exp[x]/2]*Sum[Exp[Binomial[n+1, 2]*x]/n!, {n, 0, 3*max }], {x, 0, max}] // Normal; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n] // Round, {n, 0, max} ] (* Jean-François Alcover, Apr 23 2014, after Vladeta Jovovic *) sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; Table[Length[mps[Ceiling[Range[1/2, n, 1/2]]]], {n, 5}] (* Gus Wiseman, Jul 18 2018 *) CROSSREFS Row n=2 of A219727. - Alois P. Heinz, Nov 26 2012 Cf. A007716, A007717, A014500, A014501, A020554, A094574, A316974. See also A322764. Row 0 of the array in A322765. Main diagonal of A346500. Cf. A001055, A002110. Sequence in context: A331817 A118804 A365995 * A243281 A091795 A319286 Adjacent sequences: A020552 A020553 A020554 * A020556 A020557 A020558 KEYWORD nonn AUTHOR Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe, N. J. A. Sloane 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.

Last modified December 9 15:36 EST 2023. Contains 367693 sequences. (Running on oeis4.)