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

Offset

1,4

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. Luczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.

S. Janson, D. E. Knuth, T. Luczak 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.

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

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

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. *)

Prog

(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

Crossrefs

A diagonal of A343088.

Cf. A000169, A000272, A057500, A061541, A061542, A061543, A096117, A061544, A096150, A096224.

Sequence in context: A082405 A183595 A054653 * A173370 A159307 A284768

Adjacent sequences: A061537 A061538 A061539 * A061541 A061542 A061543

Keyword

easy,nice,nonn,changed

Author

RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001

Status

approved