0,4

a(n) = number of labeled series-reduced rooted trees with n leaves (root has degree 0 or >= 2); a(n-1) = number of labeled series-reduced trees with n leaves. Also number of series-parallel networks with n labeled edges, divided by 2.

Polynomials with coefficients in triangle A008517, evaluated at 2. - Ralf Stephan, Dec 13 2004

Row sums of unsigned A134685. [From Tom Copeland (tcjpn(AT)msn.com), Oct 11 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

L. Carlitz and J. Riordan, The number of labeled two-terminal series-parallel networks, Duke Math. J. 23 (1956), 435-445 (the sequence called {A_n}).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 224.

B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.

L. R. Foulds and R. W. Robinson, Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045

Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 197.

J. Riordan, The blossoming of Schroeder's fourth problem, Acta Math., 137 (1976), 1-16.

E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.5, Equation (5.27). See also the Notes on page 66.

N. J. A. Sloane, Table of n, a(n) for n = 0..375 [Shortened file because terms grow rapidly: see Sloane link below for additional terms]

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

S. R. Finch, Series-parallel networks

Philippe Flajolet, A Problem in Statistical Classification Theory

Daniel L. Geisler, Combinatorics of Iterated Functions

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 69

N. J. A. Sloane, Table of n, a(n) for n = 0..500

Index entries for sequences related to trees

Index entries for sequences related to rooted trees

Index entries for "core" sequences

Index entries for sequences mentioned in Moon (1987)

Index entries for sequences related to parenthesizing

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 129

E.g.f. A(x) satisfies exp A(x) = 2 A(x) - x + 1.

a(0)=0, a(1)=a(2)=1; for n >= 2, a(n+1) = (n+2)*a(n) + 2*Sum_{k=2..n-1} binomial(n, k)*a(k)*a(n-k+1).

From the umbral operator L in A135494 acting on x^n comes, umbrally, (a(.) + x)^n = (n * x^(n-1) / 2) - (x^n / 2) + sum(j=1,...) [ j^(j-1) * 2^(-j) / j! * exp(-j/2) * (x+j/2)^n ] giving a(n) = 2^(-n) * sum(j=1,...) { j^(n-1) * [ (j/2) * exp(-1/2) ]^j / j! } for n > 1. - Tom Copeland (tcjpn(AT)msn.com), Feb 11 2008

M:=499; a:=array(0..500); a[0]:=0; a[1]:=1; a[2]:=1; for n from 0 to 2 do lprint(n, a[n]); od: for n from 2 to M do a[n+1]:=(n+2)*a[n]+2*add(binomial(n, k)*a[k]*a[n-k+1], k=2..n-1); lprint(n+1, a[n+1]); od:

Order := 50; t1 := solve(series((exp(A)-2*A-1), A)=-x, A); A000311 := n-> n!*coeff(t1, x, n);

(PARI) a(n)=local(A=0); if(n<0, 0, for(i=1, n, A=Pol(exp(A+x*O(x^i))-A+x-1)); n!*polcoeff(A, n))

Cf. A000669, A001003, A007827, A005804, A005805, A006351, A000084.

For n >= 2, A000311(n) = A006351(n)/2 = A005640(n)/2^(n+1).

Cf. A000110, A000669 = unlabeled hierarchies, A119649.

Row sums of A064060.

Sequence in context: A000310 A054360 A124824 this_sequence A001863 A115416 A052577

Adjacent sequences: A000308 A000309 A000310 this_sequence A000312 A000313 A000314

nonn,core,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).