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A343088 Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.

%I A343088 #19 Aug 03 2022 08:55:20
%S A343088 1,0,1,0,0,3,0,0,1,16,0,0,0,15,125,0,0,0,6,222,1296,0,0,0,1,205,3660,
%T A343088 16807,0,0,0,0,120,5700,68295,262144,0,0,0,0,45,6165,156555,1436568,
%U A343088 4782969,0,0,0,0,10,4945,258125,4483360,33779340,100000000
%N A343088 Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.
%H A343088 Andrew Howroyd, <a href="/A343088/b343088.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A343088 Triangle begins:
%e A343088   1;
%e A343088   0, 1;
%e A343088   0, 0, 3;
%e A343088   0, 0, 1, 16;
%e A343088   0, 0, 0, 15, 125;
%e A343088   0, 0, 0,  6, 222, 1296;
%e A343088   0, 0, 0,  1, 205, 3660,  16807;
%e A343088   0, 0, 0,  0, 120, 5700,  68295,  262144;
%e A343088   0, 0, 0,  0,  45, 6165, 156555, 1436568, 4782969;
%e A343088   ...
%t A343088 row[n_] := (SeriesCoefficient[#, {y, 0, n}]& /@ CoefficientList[ Log[Sum[x^k*(1+y)^Binomial[k, 2]/k!, {k, 0, n+1}]] + O[x]^(n+2), x]* Range[0, n+1]!) // Rest;
%t A343088 Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Aug 03 2022, after _Andrew Howroyd_ *)
%o A343088 (PARI)
%o A343088 Row(n)={Vec(serlaplace(polcoef(log(O(x^2*x^n)+sum(k=0, n+1, x^k*(1 + y + O(y*y^n))^binomial(k, 2)/k!)), n, y)), -(n+1))}
%o A343088 { for(n=0, 8, print(Row(n))) }
%Y A343088 Main diagonal is A000272.
%Y A343088 Subsequent diagonals give the number of connected labeled graphs with n nodes and n+k edges for k=0..11: A057500, A061540, A061541, A061542, A061543, A096117, A061544 A096150, A096224, A182294, A182295, A182371.
%Y A343088 Row sums are A322137.
%Y A343088 Column sums are A001187.
%Y A343088 Cf. A054923 (unlabeled), A062734 (transpose), A290776 (multigraphs), A322147 (loops allowed), A331437 (series-reduced).
%K A343088 nonn,tabl
%O A343088 0,6
%A A343088 _Andrew Howroyd_, Apr 14 2021