%I #22 Jan 07 2021 06:41:06

%S 1,1,1,4,26,548,22504,1708336,241874928,65285161232,34305887955616,

%T 35573982726480064,73308270568902715136,301210456065963448091072,

%U 2471487759846321319412778624,40526856087731237340916330352896,1328570640536613080046570271722309632

%N Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

%H Andrew Howroyd, <a href="/A322395/b322395.txt">Table of n, a(n) for n = 0..50</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphBridge.html">Graph Bridge</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Endpoint.html">Endpoint</a>

%F a(n) = n + Sum_{k=1..n} binomial(n,k)*A095983(k)*k^(n-k) for n >= 3. - _Andrew Howroyd_, Dec 07 2018

%t nmax = 16;

%t seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n + 1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k - 1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];

%t A095983 = seq[nmax];

%t a[n_] := If[n<3, 1, n+Sum[Binomial[n, k]*A095983[[k+1]]*k^(n-k), {k, 1, n}]];

%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Jan 07 2021, after _Andrew Howroyd_ *)

%Y Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322387, A322394.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 06 2018

%E a(6)-a(16) from _Andrew Howroyd_, Dec 07 2018