A003514 Number of series-reduced labeled graphs with n nodes.

1, 1, 2, 4, 15, 102, 4166, 402631, 76374899, 27231987762, 18177070202320, 22801993267433275, 54212469444212172845, 246812697326518127351384, 2173787304796735262709419350, 37373588848096468764431235680525, 1263513534110606141026676778422031561

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..80
• D. M. Jackson and J. W. Reilly, The enumeration of homeomorphically irreducible labeled graphs, J. Combin. Theory, B 19 (1975), 272-286.

Crossrefs

Row sums of A060514 and A307806.

The unlabeled version is A005637.

Cf. A003515 (connected).

Programs

Mathematica

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 *)

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Mar 23 2001