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A275421 Triangle read by rows: T(n,k) = number of graphs with n edges and k connected components.

%I A275421 #24 May 09 2019 09:52:26
%S A275421 1,1,1,3,1,1,5,4,1,1,12,8,4,1,1,30,23,9,4,1,1,79,57,26,9,4,1,1,227,
%T A275421 160,68,27,9,4,1,1,710,456,197,71,27,9,4,1,1,2322,1402,567,208,72,27,
%U A275421 9,4,1,1,8071,4468,1748,604,211,72,27,9,4,1,1,29503,15071,5555,1874
%N A275421 Triangle read by rows: T(n,k) = number of graphs with n edges and k connected components.
%C A275421 Multiset transformation of A002905.
%H A275421 Alois P. Heinz, <a href="/A275421/b275421.txt">Rows n = 1..60, flattened</a>
%H A275421 Peter Steinbach, <a href="/A000664/a000664_1.pdf">Field Guide to Simple Graphs, Volume 4</a>, Table 1.1a, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A275421 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A275421 T(n,1) = A002905(n).
%F A275421 T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1<k<=n.
%F A275421 G.f.: Product_{j>=1} (1-y*x^j)^(-A002905(j)). - _Alois P. Heinz_, Apr 13 2017
%e A275421       1
%e A275421       1     1
%e A275421       3     1     1
%e A275421       5     4     1     1
%e A275421      12     8     4     1     1
%e A275421      30    23     9     4     1     1
%e A275421      79    57    26     9     4     1     1
%e A275421     227   160    68    27     9     4     1     1
%e A275421     710   456   197    71    27     9     4     1     1
%e A275421    2322  1402   567   208    72    27     9     4     1     1
%e A275421    8071  4468  1748   604   211    72    27     9     4     1     1
%e A275421   29503 15071  5555  1874   615   212    72    27     9     4     1
%t A275421 rows = 12;
%t A275421 A002905 = Import["https://oeis.org/A002905/b002905.txt", "Table"][[All, 2]];
%t A275421 gf = Product[(1 - y x^j)^-A002905[[j+1]], {j, 1, rows}];
%t A275421 Rest[CoefficientList[#, y]]& /@ Rest[CoefficientList[gf + O[x]^(rows+1), x]] // Flatten (* _Jean-François Alcover_, May 09 2019, after _Alois P. Heinz_ *)
%Y A275421 Cf. A002905 (column 1), A000664 (row sums).
%K A275421 nonn,tabl
%O A275421 1,4
%A A275421 _R. J. Mathar_, Jul 27 2016