A322395 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.

1, 1, 1, 4, 26, 548, 22504, 1708336, 241874928, 65285161232, 34305887955616, 35573982726480064, 73308270568902715136, 301210456065963448091072, 2471487759846321319412778624, 40526856087731237340916330352896, 1328570640536613080046570271722309632

Offset: 0

Author

Gus Wiseman, Dec 06 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• Eric Weisstein's World of Mathematics, Graph Bridge
• Eric Weisstein's World of Mathematics, Endpoint

Crossrefs

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

Programs

Mathematica

nmax = 16;
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]]];
A095983 = seq[nmax];
a[n_] := If[n<3, 1, n+Sum[Binomial[n, k]*A095983[[k+1]]*k^(n-k), {k, 1, n}]];
a /@ Range[0, nmax] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

Formula

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

Extensions

a(6)-a(16) from Andrew Howroyd, Dec 07 2018