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 A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1). (Formerly M1504) 30
 1, 1, 1, 2, 5, 16, 78, 588, 8047, 205914, 10014882, 912908876, 154636289460, 48597794716736, 28412296651708628, 31024938435794151088, 63533059372622888758054, 244916078509480823407040988, 1783406527599529094009748567708 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is also the number of unlabeled mating graphs with n nodes. A mating graph has no two vertices with identical sets of neighbors. - Tanya Khovanova, Oct 23 2008 REFERENCES F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1. F. Harary and E. Palmer, Graphical Enumeration, (1973), compare formula (8.7.11). R. W. Robinson, personal communication. R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from R. W. Robinson) David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013. Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612. R. J. Mathar, Illustrations for n=1..5 nodes Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989. R. W. Robinson, Graphs without endpoints - computer printout N. J. A. Sloane, Illustration of a(0)-a(5) MATHEMATICA permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t * k; s += t]; s!/m]; edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]]; a[n_] := Sum[permcount[p] * 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; a[0] = 1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *) PROG (PARI) \\ Compare A000088. 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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)} a(n) = {my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s} \\ Andrew Howroyd, Sep 09 2018 CROSSREFS Row sums of A123551. Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A006024 (number of labeled mating graphs with n nodes), A129584 (bi-point-determining graphs). If isolated nodes are forbidden, see A261919. Sequence in context: A263914 A218168 A054960 * A236960 A290609 A048754 Adjacent sequences:  A004107 A004108 A004109 * A004111 A004112 A004113 KEYWORD nonn AUTHOR STATUS approved

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Last modified September 18 10:51 EDT 2021. Contains 347518 sequences. (Running on oeis4.)