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OFFSET
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0,4
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COMMENTS
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"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
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REFERENCES
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Miklos Bona, 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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..50
Matthias Beck, Benjamin Braun and Nguyen Le, Mahonian partition identities via
polyhedral geometry, arXiv:1103.1070 [math.NT], 2011.
Huantian Cao, AutoGF: An Automated
System to Calculate Coefficients of Generating Functions.
Patrick De Causmaecker, Stefan De Wannemacker, On the number of antichains of sets
in a finite universe, arXiv:1407.4288 [math.CO], 2014.
P. Flajolet and R. Sedgewick, Analytic
Combinatorics, 2009; see page 138
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.
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.
R. W. Robinson, First 50 terms of A1187 and
A1188
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.
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FORMULA
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n*2^binomial(n, 2) = Sum_k binomial(n,
k)*k*a(k)*2^binomial(n-k, 2).
E.g.f.: 1 + log( sum( 2^binomial(n, 2) * x^n / n!,
n=0..infinity) ). - Michael
Somos, Jun 12 2000
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EXAMPLE
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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! + ...
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MAPLE
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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
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MATHEMATICA
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g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}];
Range[0, 20]! CoefficientList[Series[Log[g] + 1, {x, 0, 20}], 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 *)
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PROG
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(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
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CROSSREFS
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Logarithmic transform of A006125 (labeled
graphs). Cf. A053549.
Row sums of triangle A062734.
Sequence in context: A084284 A084285 A084286 * A093377 A178017 A131591
Adjacent sequences: A001184 A001185 A001186 * A001188 A001189 A001190
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N.
J. A. Sloane
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STATUS
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approved
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