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%I #35 Sep 27 2017 04:29:03
%S 1,1,1,2,1,1,6,3,1,1,21,8,3,1,1,112,30,9,3,1,1,853,145,32,9,3,1,1,
%T 11117,1028,154,33,9,3,1,1,261080,12320,1065,156,33,9,3,1,1,11716571,
%U 274806,12513,1074,157,33,9,3,1,1,1006700565,12007355,276114,12550,1076,157,33,9,3,1,1
%N Triangle read by rows: T(n,m) = number of unlabeled graphs on n nodes with m connected components, m = 1,2,...,n.
%H Alois P. Heinz, <a href="/A201922/b201922.txt">Rows n = 1..75, flattened</a>
%H P. Flajolet, R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/book.pdf">Analytic combinatorics</a>, Theorem I.1 (Multiset)
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 (2017) Table 82
%H Peter Steinbach, <a href="/A000664/a000664_5.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 5 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%F T(n,m) = sum over the partitions of n with m parts: 1*K1 + 2*K2 + ... + n*Kn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A001349(i) + Ki - 1, Ki).
%F O.g.f.: Product_{n>=1} 1/(1 - y*x^n)^A001349(n). - _Geoffrey Critzer_, Apr 19 2012
%e Triangle starts:
%e 1
%e 1 1
%e 2 1 1
%e 6 3 1 1
%e 21 8 3 1 1
%e 112 30 9 3 1 1
%e 853 145 32 9 3 1 1 ...
%t nn=10; c=(A000088=Table[NumberOfGraphs[n], {n,0,nn}]; f[x_] = 1-Product[1/(1-x^k)^a[k], {k,1,nn}]; 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); f[list_]:=Select[list,#>0&]; g=Product[1/(1-y x^n)^c[[n+1]], {n,1,nn}]; Map[f, Drop[CoefficientList[Series[g, {x,0,nn}], {x,y}],1]] //Flatten (* _Geoffrey Critzer_, Apr 19 2012 (c in above Mma code is given by Jean Francois Alcover in A001349) *)
%Y Cf. A001349 (first column), A000088 (row sum), A201968 (limits in the diagonals), A106240, A274934 (2nd column).
%K nonn,tabl,nice
%O 1,4
%A _Max Alekseyev_, Dec 06 2011