A003514 - OEIS

%I M1290 #40 Feb 24 2024 04:32:13

%S 1,1,2,4,15,102,4166,402631,76374899,27231987762,18177070202320,

%T 22801993267433275,54212469444212172845,246812697326518127351384,

%U 2173787304796735262709419350,37373588848096468764431235680525,1263513534110606141026676778422031561

%N Number of series-reduced labeled graphs with n nodes.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A003514/b003514.txt">Table of n, a(n) for n = 0..80</a>

%H D. M. Jackson and J. W. Reilly, <a href="https://doi.org/10.1016/0095-8956(75)90090-8">The enumeration of homeomorphically irreducible labeled graphs</a>, J. Combin. Theory, B 19 (1975), 272-286.

%F E.g.f.: (1 + x)^( - 1/2) * exp(x/2 - x^2/4) * Sum_{k=0..inf} (2 * exp( - x/(1 + x)))^binomial(k, 2) * (exp(x^2/2/(1 + x)))^k * x^k/k!. - _Vladeta Jovovic_, Mar 23 2001

%t max = 15; f[x_] := (1 + x)^(-1/2)*Exp[x/2-x^2/4]*Sum[(2*Exp[-x/(1+x)])^Binomial[k, 2]*Exp[x^2/2/(1+x)]^k*x^k/k!, {k, 0, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]!(* _Jean-François Alcover_, Nov 25 2011, after Vladeta Jovovic *)

%o (PARI) seq(n)={my(x='x+O('x^(n+1))); Vec(serlaplace((1 + x)^( - 1/2) * exp(x/2 - x^2/4) * sum(k=0, n, (2 * exp(-x/(1 + x)))^binomial(k, 2) * (exp(x^2/2/(1 + x)))^k * x^k/k!)))} \\ _Andrew Howroyd_, Feb 23 2024

%Y Row sums of A060514 and A307806.

%Y The unlabeled version is A005637.

%Y Cf. A003515 (connected).

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Mar 23 2001