A061540 Number of connected labeled graphs with n nodes and n+1 edges.
0, 0, 0, 6, 205, 5700, 156555, 4483360, 136368414, 4432075200, 154060613850, 5720327205120, 226378594906035, 9523895202838016, 424814409531910125, 20037831121798963200, 996964614369038858060, 52198565072252054814720, 2869621989939313379211204, 165302832533722012508160000
Offset: 1
References
- Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 407, Eq. (6.5).
Links
- Sergey Serebryakov, Table of n, a(n) for n = 1..40
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
- E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.
Crossrefs
Programs
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Maple
A001864 := proc(n) add(binomial(n,s)*s^s*(n-s)^(n-s),s=1..n-1) ; end proc: A061540 := proc(n) (n-1)*(5*n^2+3*n+2)*n^(n-2)-14*A001864(n) ; %/24 ; end proc: # R. J. Mathar, May 10 2016 see Chapter 6.3 in Bona's Handbook of Combinatorics
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Mathematica
max = 18; t[x_] := -ProductLog[-x]; w1[x_] := t[x]^4/24*(6-t[x])/(1-t[x])^3; Drop[ CoefficientList[ Series[ w1[x], {x, 0, max}], x]*Range[0, max]!, 1] (* Jean-François Alcover, Apr 02 2012, after e.g.f. *) -
Python
from math import comb def A061540(n): return 0 if n<2 else ((n*(n*(5*n - 2) - 1) - 2)*n**(n-2)-14*((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n)))//24 # Chai Wah Wu, Apr 26 2023
Formula
E.g.f.: W1(x) := T(x)^4/24 * (6-T(x))/(1-T(x))^3 where T(x) is the e.g.f. for rooted labeled trees (A000169), i.e. T(x) = -LambertW(-x) = x*exp(T(x)).
a(n) ~ 5*n^(n+1)/24 * (1 - 7/5*sqrt(2*Pi/n)). - Vaclav Kotesovec, Jul 09 2013