OFFSET
1,2
REFERENCES
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 Problem 5.39 (which deals with the labeled case of the same sequence).
LINKS
Jean-François Alcover, Table of n, a(n) for n = 1..100
B. I. Bayoumi, M. H. El-Zahar and S. M. Khamis, Asymptotic enumeration of N-free partial orders, Order 6 (1989), 219-232.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.
Frédéric Fauvet, L. Foissy, D. Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149 [math.CO], 2016.
S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72.
Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
R. P. Stanley, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299. Math. Rev. 50 #4416.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
FORMULA
G.f. A(x) = 1 + x + 2*x^2 + 5*x^3 + ... satisfies A(x) = exp(Sum_{k>=1} (1/k)*(A(x^k) + 1/A(x^k) - 2 + x^k)).
From: Andrew Howroyd, Nov 26 2020: (Start)
Euler transform of A007453.
G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A007454.
(End)
EXAMPLE
From Andrew Howroyd, Nov 26 2020: (Start)
In the following examples of series-parallel networks, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 2: (oo), (o|o).
a(3) = 5: (ooo), (o(o|o)), ((o|o)o), (o|o|o), (o|oo).
a(4) = 15: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)oo), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)), ((o|oo)o), ((o|o|o)o), (o|o|o|o), (o|o|oo), (oo|oo), (o|(ooo)), (o|(o(o|o))), (o|((o|o)o)).
(End)
MATHEMATICA
terms = 25; A[_] = 1; Do[A[x_] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms + 1}]] + O[x]^(terms + 1) // Normal, terms + 1];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jun 29 2011, updated Jan 12 2018 *)
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
Name corrected by Salah Uddin Mohammad, Jun 07 2020
STATUS
editing