%I #28 Aug 16 2019 07:49:52
%S 1,1,0,1,10,253,12058,1052443,169488200,51045018089,29184193354806,
%T 32122530765469967,68867427921051098084,290155706369032525823085,
%U 2417761578629525173499004146,40013923790443379076988789688611,1318910080173114018084245406769861936
%N Number of n-node connected labeled graphs without endpoints.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 404.
%H Vaclav Kotesovec, <a href="/A059166/b059166.txt">Table of n, a(n) for n = 0..80</a>
%F a(n) = Sum_{i=0..n} (-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, for n>2, a(0)=1, a(1)=1, a(2)=0, where c(n) is number of n-node connected labeled graphs (cf. A001187).
%F E.g.f.: 1 + x^2/2 + log(Sum_{n >= 0} 2^binomial(n, 2)*(x*exp(-x))^n/n!).
%F a(n) ~ 2^(n*(n-1)/2). - _Vaclav Kotesovec_, May 14 2015
%F Logarithmic transform of A100743, if we assume a(1) = 0. - _Gus Wiseman_, Aug 15 2019
%p c:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
%p add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*c(k), k=1..n-1)/n)
%p end:
%p a:= n-> max(0, add((-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, i=0..n)):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Oct 27 2017
%t Flatten[{1,1,0,Table[n!*Sum[(-1)^(n-j)*SeriesCoefficient[1+Log[Sum[2^(k*(k-1)/2)*x^k/k!,{k,0,j}]],{x,0,j}]*j^(n-j)/(n-j)!,{j,0,n}],{n,3,15}]}] (* _Vaclav Kotesovec_, May 14 2015 *)
%t c[0] = 1; c[n_] := c[n] = 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k)*(n - k - 1)/2)*c[k], {k, 1, n-1}]/n; a[0] = a[1] = 1; a[2] = 0; a[n_] := Sum[(-1)^i*Binomial[n, i]*c[n-i]*(n-i)^i, {i, 0, n}]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Oct 27 2017, using _Alois P. Heinz_'s code for c(n) *)
%o (PARI) seq(n)={Vec(serlaplace(1 + x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))))} \\ _Andrew Howroyd_, Sep 09 2018
%Y Cf. A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).
%Y Cf. A001187, A007146, A095983, A100743, A261919, A322395.
%K easy,nonn
%O 0,5
%A _Vladeta Jovovic_, Jan 12 2001
%E More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001