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A059167 Number of n-node labeled graphs without endpoints.

%I A059167 #26 Feb 07 2024 19:47:55
%S A059167 1,1,1,2,15,314,13757,1142968,178281041,52610850316,29702573255587,
%T A059167 32446427369694348,69254848513798160815,291053505824567573585744,
%U A059167 2421830049319361003822380177,40050220743831370293688592267252,1319550593412053164173947687592553185
%N A059167 Number of n-node labeled graphs without endpoints.
%D A059167 F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).
%H A059167 Robert Israel, <a href="/A059167/b059167.txt">Table of n, a(n) for n = 0..81</a>
%H A059167 Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, <a href="http://math.stackexchange.com/questions/1930410/">Proof of the closed form of the e.g.f. by combinatorial species</a>. - _Marko Riedel_, Sep 18 2016
%F A059167 a(n) = Sum_{i=0..n-1} binomial(n-1, i)*b(i+1)*a(n-i-1), n>0, a(0)=1, where b(n) is number of n-node connected labeled graphs without endpoints (Cf. A059166).
%F A059167 E.g.f.: exp(x^2/2)*(Sum_{n >= 0} 2^binomial(n, 2)*(x/exp(x))^n/n!). - _Vladeta Jovovic_, Mar 23 2004
%F A059167 a(n) ~ 2^(n*(n-1)/2). - _Vaclav Kotesovec_, Sep 22 2016
%p A059167 F:= proc(N) local S;
%p A059167    S:= series(exp(1/2*x^2)*Sum(2^binomial(n, 2)*(x/exp(x))^n/n!, n = 0 .. N),x,N+1);
%p A059167    seq(coeff(S,x,i)*i!,i=0..N)
%p A059167 end proc:
%p A059167 F(20); # _Robert Israel_, Sep 18 2016
%t A059167 b[n_] := If[n < 3, Boole[n == 1], 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}]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, i] b[i + 1] a[n - i - 1], {i, 0, n - 1}]; Table[a@ n, {n, 0, 15}] (* _Michael De Vlieger_, Sep 19 2016, after _Vaclav Kotesovec_ at A059166 *)
%o A059167 (PARI) seq(n)={my(A=x/exp(x + O(x^n))); Vec(serlaplace(exp(x^2/2 + O(x*x^n)) * sum(k=0, n, 2^binomial(k, 2)*A^k/k!)))} \\ _Andrew Howroyd_, Sep 09 2018
%Y A059167 Column k=0 of A327369.
%Y A059167 Cf. A059166 (n-node connected labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).
%K A059167 easy,nonn
%O A059167 0,4
%A A059167 _Vladeta Jovovic_, Jan 12 2001
%E A059167 More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001