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A339063 Number of unlabeled simple graphs with n edges rooted at two noninterchangeable vertices. 9
1, 4, 13, 43, 141, 467, 1588, 5544, 19966, 74344, 286395, 1141611, 4707358, 20063872, 88312177, 400980431, 1875954361, 9032585846, 44709095467, 227245218669, 1184822316447, 6330552351751, 34630331194626, 193785391735685, 1108363501628097, 6474568765976164 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

EXAMPLE

The a(1) = 4 cases correspond to a single edge which can be attached to zero, one or both of the roots.

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_, t_] := Product[With[{g = GCD[v[[i]], v[[j]]]}, t[v[[i]]*v[[j]]/ g]^g], {i, 2, Length[v]}, {j, 1, i-1}]*Product[With[{c = v[[i]]}, t[c]^Quotient[c-1, 2]*If[OddQ[c], 1, t[c/2]]], {i, 2, Length[v]}];

G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 1+x^#&], {p, IntegerPartitions[n]}]; s/n!];

seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]//CoefficientList[#, x]&];

seq[15] (* Jean-Fran├žois Alcover, Dec 03 2020, after Andrew Howroyd *)

PROG

(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, 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)))}

G(n, x, r)={my(s=0); forpart(p=n, s+=permcount(p)*edges(concat(r, Vec(p)), i->1+x^i)); s/n!}

seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}

CROSSREFS

Cf. A000664, A053419 (one root), A304070, A339040, A339064, A339065.

Sequence in context: A072307 A266494 A121486 * A188176 A003688 A033434

Adjacent sequences:  A339060 A339061 A339062 * A339064 A339065 A339066

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Nov 22 2020

STATUS

approved

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Last modified April 13 03:43 EDT 2021. Contains 342934 sequences. (Running on oeis4.)