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A126100
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Number of rooted connected unlabeled graphs on n nodes.
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6
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0, 1, 1, 3, 11, 58, 407, 4306, 72489, 2111013, 111172234, 10798144310, 1944301471861, 650202565436890, 404697467417019634, 470133531223369393920, 1022561022228933341815171, 4177761667636803276899047351, 32163582481439081597751699343141, 468019937132164016636736323752098741
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OFFSET
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0,4
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COMMENTS
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Let G run through all connected unlabeled graphs on n nodes. Add up the numbers of inequivalent nodes (under Aut(G)) for each G.
"Pointed" connected graphs. This has the same relation to A001349 as A000081 does to A000055.
a(0) = 0 because the empty graph cannot be rooted.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from David Applegate and N. J. A. Sloane)
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FORMULA
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The g.f. A(x) = x+x^2+3*x^3+11*x^4+... satisfies f(x) = 1 + A(x)*g(x), where f(x) = 1+x+2*x^2+6*x^3+20*x^4+... is the g.f. for A000666 and g(x) = 1+x+2*x^2+4*x^3+11*x^4+... is the g.f. for A000088. - Brendan McKay
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EXAMPLE
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For 3 nodes G is either a path (2 kinds of nodes) or a triangle (one kind of node), for a total of a(3) = 3.
For the 5-vertex graphs we have 2 x 1 orbit, 6 x 2 orbits, 8 x 3 orbits, 5 x 4 orbits for a total of 2 + 12 + 24 + 20 = 58.
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MATHEMATICA
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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]];
g[n_, r_] := (s = 0; Do[s += permcount[p]*(2^(r*Length[p] + edges[p])), {p, IntegerPartitions[n]}]; s/n!);
seq[m_] := Sum[g[n-1, 1] x^(n-1), {n, 0, m}]/Sum[g[n-1, 0] x^(n-1), {n, 0, m}] + O[x]^m // CoefficientList[#, x]& // Prepend[#, 0]&;
seq[20] (* Jean-François Alcover, Jul 09 2018, after Andrew Howroyd *)
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PROG
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(PARI)
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)}
g(n, r) = {my(s=0); forpart(p=n, s+=permcount(p)*(2^(r*#p+edges(p)))); s/n!}
seq(n)={concat([0], Vec(Ser(vector(n, n, g(n-1, 1)))/Ser(vector(n, n, g(n-1, 0)))))} \\ Andrew Howroyd, May 03 2018
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CROSSREFS
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Cf. A001349, A126101, A000666, A000088, A126201, A303831 (birooted), A304311.
Sequence in context: A229512 A208990 A020012 * A009444 A305990 A168325
Adjacent sequences: A126097 A126098 A126099 * A126101 A126102 A126103
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KEYWORD
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nonn,nice
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AUTHOR
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David Applegate and N. J. A. Sloane, Mar 05 2007
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EXTENSIONS
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a(5)-a(9) computed by Gordon F. Royle, Mar 05 2007
a(10) and a(11) computed by Brendan McKay, Mar 05 2007
a(12) onwards computed from the generating function, A000088 and A000666 by David Applegate and N. J. A. Sloane, Mar 06 2007
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STATUS
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approved
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