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%I #9 Feb 03 2024 11:01:45
%S 1,2,2,4,8,17,39,92,227,573,1482,3883,10343,27786,75392,205933,566166,
%T 1564316,4342431,12100382,33836606,94903889,266914438,752517020,
%U 2126292931,6020035120,17075411671,48514471709,138051863755,393397897262,1122523343690
%N Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with at most n edges.
%C The graphs considered here can have loops but not parallel edges.
%H Andrew Howroyd, <a href="/A369289/b369289.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = A000055(n) + A368983(n) = A000055(n) + A000081(n) + A001429(n) for n > 0.
%F Inverse Euler transform of A369145.
%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 + 2*g(1) - 2*g(1)^2 )/2) }
%Y Cf. A000055, A000081, A001429, A368983, A369145.
%K nonn
%O 0,2
%A _Andrew Howroyd_, Feb 02 2024
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