A339065
Number of unlabeled loopless multigraphs with n edges rooted at two noninterchangeable vertices.
Original entry on oeis.org
1, 4, 17, 69, 281, 1147, 4784, 20345, 88726, 396971, 1823920, 8605364, 41684417, 207201343, 1056244832, 5518054182, 29521703655, 161625956908, 904857279576, 5176569819167, 30241443710950, 180293374961036, 1096240011165724, 6793998104717138, 42894087222036022, 275735424352928682
Offset: 0
The a(1) = 4 cases correspond to a single edge which can be attached to zero, one or both of the roots.
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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_, 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, 1, Length[v]}];
G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 1/(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-François Alcover, Dec 01 2020, after Andrew Howroyd *)
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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/(1-x^i))); s/n!}
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}
A339036
Number of unlabeled connected loopless multigraphs with n edges rooted at one distinguished vertex.
Original entry on oeis.org
1, 1, 3, 9, 30, 104, 390, 1518, 6208, 26372, 116221, 529341, 2487054, 12027502, 59778867, 304916272, 1594273763, 8535706749, 46753269749, 261771468438, 1497087288210, 8739579074131, 52045067963540, 315980654042243, 1954770128712348, 12315770916526091
Offset: 0
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seq[n_] := G[2n, x+O[x]^n, {1}]/G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}
A339037
Number of unlabeled connected loopless multigraphs with n edges rooted at one oriented edge.
Original entry on oeis.org
1, 3, 11, 41, 160, 641, 2672, 11479, 50938, 232830, 1095151, 5292990, 26257328, 133548307, 695752146, 3709509938, 20224607541, 112675185837, 641016837378, 3721624588590, 22037618432547, 133023405207408, 818085097509494, 5123460267381837, 32660335570381961, 211825198708110059
Offset: 1
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seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]/G[2n, x+A, {}] // CoefficientList[#, x]&]; (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1,1])/G(2*n, x+A, []))}
A339040
Number of unlabeled connected simple graphs with n edges rooted at two noninterchangeable vertices.
Original entry on oeis.org
1, 3, 10, 35, 125, 460, 1747, 6830, 27502, 113987, 485971, 2129956, 9591009, 44341610, 210345962, 1023182861, 5100235807, 26035673051, 136023990102, 726877123975, 3970461069738, 22156281667277, 126234185382902, 733899631974167, 4351500789211840
Offset: 1
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, [])); Vec(G(2*n, x+A, [1, 1])/g - (G(2*n, x+A, [1])/g)^2)}
A339043
Number of unlabeled connected loopless multigraphs with n edges rooted at two indistinguishable vertices.
Original entry on oeis.org
1, 3, 11, 43, 178, 767, 3425, 15783, 74775, 363639, 1811808, 9239430, 48175945, 256658465, 1396152633, 7750325528, 43882706171, 253308596926, 1490040961732, 8928063141435, 54469529215562, 338236254603888, 2136952452531537, 13731571816349732, 89710429044324926
Offset: 1
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seq[n_] := Module[{g, gr}, g = G[2n, x+O[x]^n, {}]; gr = G[2n, x+O[x]^n, {1}]/g; G[2n, x+O[x]^n, {1, 1}]/g - gr^2 + G[2n, x+O[x]^n, {2}]/g - (Normal[gr] /. x -> x^2) // CoefficientList[#/2, x]& // Rest];
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, []), gr=G(2*n, x+A, [1])/g); Vec(G(2*n, x+A, [1, 1])/g - gr^2 + G(2*n, x+A, [2])/g - subst(gr, x, x^2))/2}
A339038
Number of unlabeled connected loopless multigraphs with n edges rooted at one unoriented edge.
Original entry on oeis.org
1, 2, 7, 23, 88, 339, 1396, 5915, 26080, 118539, 555678, 2678458, 13262193, 67353325, 350493424, 1866989802, 10171394388, 56631507822, 322011612423, 1868702977253, 11061267210030, 66745602611831, 410360493588788, 2569318971123439, 16374787277199728, 106180292431149021
Offset: 1
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seq[n_] := (G[2n, x + O[x]^n, {1, 1}] + G[2n, x + O[x]^n, {2}])/G[2n, x + O[x]^n, {}] // CoefficientList[#/2, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1,1]) + G(2*n, x+A, [2]))/G(2*n, x+A, []))/2}
Showing 1-6 of 6 results.