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A001187 Number of connected labeled graphs with n nodes.
(Formerly M3671 N1496)
145
1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896, 1328578958335783201008338986845427712 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
"Based on experimental data obtained using the software LattE [14] and the Online Encyclopedia of Integer Sequences [19], we make the following conjecture: Conjecture 11. For j >= 2, Vol(C_j ) is equal to the number of labeled connected graphs on j - 1 vertices." [Beck et al., 2011]
For n > 1, a(n) is log-convex. Furthermore, a(n+1)*a(n-1) ~ 2*a(n)*a(n). - Ran Pan, Oct 28 2015
a(n) is also the number of tournaments on {1,...,n} for which 1 is reachable from every vertex. - Don Knuth, Aug 06 2020
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 398-402.
D. G. Cantor, personal communication.
Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 518.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.1.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 78.
LINKS
Matthias Beck, Benjamin Braun and Nguyen Le, Mahonian partition identities via polyhedral geometry, arXiv:1103.1070 [math.NT], 2011.
Marco Coraggio and Mario di Bernardo, Data-driven design of complex network structures to promote synchronization, arXiv:2309.10941 [eess.SY], 2023.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 138.
Philippe Fournier Viger, Ganghuan He, Chun-Wei Jerry Lin, and Heitor Murilo Gomes, Mining Attribute Evolution Rules in Dynamic Attributed Graphs, Proceedings of the Big Data Analytics and Knowledge Discovery, 22nd International Conference, (Bratislava, Slovakia, DaWaK 2020).
Adriano M. Garsia, James Haglund, Dun Qiu, and Marino Romero, e-Positivity Results and Conjectures, arXiv:1904.07912 [math.CO], 2019.
E. N. Gilbert, Enumeration of labelled graphs, Canad. J. Math., 8 (1956), 405-411.
E. N. Gilbert, Enumeration of labelled graphs (Annotated scanned copy)
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91.
Arun P. Mani and R. J. Stones, The Number of Labeled Connected Graphs Modulo Prime Powers, SIAM Journal on Discrete Mathematics, Vol. 30, No. 2, pp. 1046-1057.
Alexey A. Melnikov, Leonid E. Fedichkin, Ray-Kuang Lee, and Alexander Alodjants, Machine learning transfer efficiencies for noisy quantum walks, arXiv:2001.05472 [quant-ph], 2020. Also in Advanced Quantum Technologies (2020) Vol. 3, 1900115.
Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074)
J. Novak, Three lectures on free probability, arXiv preprint arXiv:1205.2097 [math.CO], 2012.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Labeled Graph
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 87, Eq. 3.10.2.
FORMULA
n * 2^binomial(n, 2) = Sum_{k=1..n} binomial(n, k) * k * a(k) * 2^binomial(n-k, 2).
E.g.f.: 1 + log(Sum_{n>=0} 2^binomial(n, 2) * x^n / n!). - Michael Somos, Jun 12 2000
EXAMPLE
E.g.f.: 1 + x + x^2/2! + 4*x^3/3! + 38*x^4/4! + 728*x^5/5! + 26704*x^6/6! + 1866256*x^7/7! + 251548592*x^8/8! + ...
MAPLE
t1 := 1+log( add(2^binomial(n, 2)*x^n/n!, n=0..30)): t2 := series(t1, x, 30): A001187 := n->n!*coeff(t2, x, n);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*a(k), k=1..n-1)/n)
end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 26 2013
# Alternative:
a := proc(n) option remember;
2^((n-1)*n/2) - add(binomial(n-1, k)*2^((k-n+1)*(k-n+2)/2)*a(k+1), k=0..n-2)
end:
seq(a(n), n=0..16); # Peter Luschny, Feb 21 2023
MATHEMATICA
m:=20; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, m}]; Range[0, m]! CoefficientList[Series[Log[g] +1, {x, 0, m}], x] (* Geoffrey Critzer, Nov 12 2011 *)
a[n_]:= a[n]= If[n==0, 1, 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]* 2^((n-k)*(n-k-1)/2)*a[k], {k, 1, n-1}]/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Log[ Sum[2^(k(k-1)/2) x^k/k!, {k, 0, n}]], {x, 0, n}]]; (* Michael Somos, Jul 11 2019 *)
PROG
(PARI) {a(n) = if(n<0, 0, n! * polcoeff( 1 + log( sum( k=0, n, 2^binomial(k, 2) * x^k / k!, x * O(x^n))), n))}; /* Michael Somos, Jun 12 2000 */
(Sage)
@cached_function
def A001187(n):
if n == 0: return 1
return 2^(n*(n-1)/2)- sum(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*A001187(k) for k in (1..n-1))/n
[A001187(n) for n in (0..15)] # Peter Luschny, Jan 17 2016
(Magma)
m:=30;
f:= func< x | 1+Log( (&+[2^Binomial(n, 2)*x^n/Factorial(n): n in [0..m+3]]) ) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022
CROSSREFS
Logarithmic transform of A006125 (labeled graphs).
Row sums of triangle A062734.
Cf. A053549.
Sequence in context: A084284 A084285 A084286 * A093377 A178017 A131591
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)