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A275421 Triangle read by rows: T(n,k) = number of graphs with n edges and k connected components. 12

%I

%S 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 160,68,27,9,4,1,1,710,456,197,71,27,9,4,1,1,2322,1402,567,208,72,27,

%U 9,4,1,1,8071,4468,1748,604,211,72,27,9,4,1,1,29503,15071,5555,1874

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

%C Multiset transformation of A002905.

%H Alois P. Heinz, <a href="/A275421/b275421.txt">Rows n = 1..60, flattened</a>

%H 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 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>

%F T(n,1) = A002905(n).

%F 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 G.f.: Product_{j>=1} (1-y*x^j)^(-A002905(j)). - _Alois P. Heinz_, Apr 13 2017

%e 1

%e 1 1

%e 3 1 1

%e 5 4 1 1

%e 12 8 4 1 1

%e 30 23 9 4 1 1

%e 79 57 26 9 4 1 1

%e 227 160 68 27 9 4 1 1

%e 710 456 197 71 27 9 4 1 1

%e 2322 1402 567 208 72 27 9 4 1 1

%e 8071 4468 1748 604 211 72 27 9 4 1 1

%e 29503 15071 5555 1874 615 212 72 27 9 4 1

%t rows = 12;

%t A002905 = Import["https://oeis.org/A002905/b002905.txt", "Table"][[All, 2]];

%t gf = Product[(1 - y x^j)^-A002905[[j+1]], {j, 1, rows}];

%t Rest[CoefficientList[#, y]]& /@ Rest[CoefficientList[gf + O[x]^(rows+1), x]] // Flatten (* _Jean-Fran├žois Alcover_, May 09 2019, after _Alois P. Heinz_ *)

%Y Cf. A002905 (column 1), A000664 (row sums).

%K nonn,tabl

%O 1,4

%A _R. J. Mathar_, Jul 27 2016

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Last modified February 23 11:21 EST 2020. Contains 332159 sequences. (Running on oeis4.)