A303832
The number of edge-rooted unlabeled connected graphs with n edges.
Original entry on oeis.org
1, 1, 4, 10, 32, 101, 346, 1220, 4517, 17338, 69107, 285009, 1215015, 5344224, 24223641, 113001129, 541913075, 2668817544, 13484234188, 69831773559, 370361639587, 2009988998148, 11153858854425, 63242354288220, 366140089188603, 2163036956456422, 13031489297543608
Offset: 1
a(1)=1: the connected graph with 1 edge (which is rooted).
a(2)=1: the connected graph with 2 edges (one rooted).
a(3)=4: the triangle graph with one choice of rooting, the linear tree with either the middle or a terminating edge rooted, the star graph with one edge rooted.
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/(2*G(2*n, x+A, [])*(1+x)))} \\ Andrew Howroyd, Nov 21 2020
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}
A339039
Number of unlabeled connected simple graphs with n edges rooted at one distinguished vertex.
Original entry on oeis.org
1, 1, 2, 5, 13, 37, 114, 367, 1248, 4446, 16526, 63914, 256642, 1067388, 4590201, 20376849, 93240065, 439190047, 2126970482, 10579017047, 53983000003, 282345671127, 1512273916781, 8287870474339, 46438619162441, 265840311066579
Offset: 0
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}
A339044
Number of unlabeled connected simple graphs with n edges rooted at one oriented edge.
Original entry on oeis.org
1, 2, 6, 18, 57, 188, 651, 2336, 8719, 33741, 135185, 559908, 2394326, 10557283, 47943126, 223987316, 1075455181, 5301593544, 26807904317, 138924912857, 737220195148, 4002876571636, 22221898966507, 126042573704637, 729944250603862, 4313430995825272
Offset: 1
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1, 1])/G(2*n, x+A, [])/(1+x))}
A339064
Number of unlabeled simple graphs with n edges rooted at two indistinguishable vertices.
Original entry on oeis.org
1, 3, 9, 28, 87, 276, 909, 3086, 10879, 39821, 151363, 597062, 2442044, 10342904, 45301072, 204895366, 955661003, 4590214994, 22675644514, 115068710553, 599149303234, 3197694533771, 17475917252052, 97712883807625, 558481251055893, 3260409769087068
Offset: 0
The a(1) = 3 cases correspond to a single edge which can be attached to zero, one or both of the roots.
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/2)}
Showing 1-6 of 6 results.