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A182117 The number of simple unlabelled graphs whose connected components have distinct sizes (no two connected components have the same number of nodes). 1

%I #13 Aug 14 2019 22:03:59

%S 1,1,1,3,8,29,141,1004,12157,273566,11992403,1018721807,165079148151,

%T 50501012085899,29053990553658291,31426435466752989418,

%U 64000986650206723016967,245935832726996458741850870,1787577661144566941500116028526,24637809007189108944263048282845188

%N The number of simple unlabelled graphs whose connected components have distinct sizes (no two connected components have the same number of nodes).

%F O.g.f.: prod(n>=1, 1 + A001349(n)*x^n ) where A001349 is the number of connected graphs.

%t nn = 19; c = (A000088 = Table[NumberOfGraphs[n], {n, 0, nn}]; f[x_] = 1 - Product[1/(1 - x^k)^a[k], {k, 1, nn}];

%t a[0] = a[1] = a[2] = 1; coes = CoefficientList[Series[f[x], {x, 0, nn}], x]; sol = First[Solve[Thread[Rest[coes + A000088] == 0]]]; Table[a[n], {n, 0, nn}] /. sol); p = Product[1 + c[[n + 1]] x^n, {n, 1, nn}]; CoefficientList[ Series[p, {x, 0, nn}], x]

%t (* The Mma code for c in the above is given by _Jean-Fran├žois Alcover_ in A001349. *)

%Y Cf. A207828.

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Apr 12 2012

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