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%I #10 Jan 11 2024 16:47:01
%S 1,1,1,3,6,14,33,81,204,526,1376,3648,9792,26485,72233,198192,546846,
%T 1515687,4218564,11782427,33013541,92759384,261290682,737688946,
%U 2086993034,5915398230,16795618221,47763406249,136028420723,387928330677,1107692471888,3166613486137
%N Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with n edges.
%C The graphs considered here can have loops but not parallel edges.
%H Andrew Howroyd, <a href="/A368983/b368983.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = A000081(n) + A001429(n) = A068051(n) - A027852(n) for n > 0.
%e Representatives of the a(3) = 3 graphs are:
%e {{1,2}, {1,3}, {2,3}},
%e {{1}, {1,2}, {1,3}},
%e {{1}, {1,2}, {2,3}}.
%o (PARI) \\ TreeGf gives gf of A000081.
%o TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2)}
%Y A diagonal of A322114.
%Y The labeled version is A368951.
%Y Cf. A000081, A001429, A027852, A068051, A283755, A368984 (not necessarily connected).
%K nonn
%O 0,4
%A _Andrew Howroyd_, Jan 11 2024
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