OFFSET
1,2
COMMENTS
For a simple relationship to series-reduced rooted trees, partitions of n, and phylogenetic trees among other combinatoric constructs, see comments in A000311. - Tom Copeland, Jan 06 2021
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 417.
P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
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 5.40(a), S(x).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..200
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4. [From Aaron Meyerowitz, Sep 30 2010]
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1 ), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
ISCGI, Cograph graphs
Song He, Fei Teng, and Yong Zhang, String Correlators: Recursive Expansion, Integration-by-Parts and Scattering Equations, arXiv:1907.06041 [hep-th], 2019. Also in Journal of High Energy Physics (2019), 2019:85.
W. Knoedel, Über Zerfällungen, Monatsh. Math., 55 (1951), 20-27.
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
M. D. Moffitt, Search Strategies for Topological Network Optimization, Proceedings of the AAAI Conference on Artificial Intelligence, 36(9) (2022), 10299-10308.
J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the e.g.f. U(x)).
N. J. A. Sloane, Transforms
Index entries for sequences related to trees [From Aaron Meyerowitz, Sep 30 2010]
FORMULA
E.g.f. is reversion of 2*log(1+x)-x.
Also exponential transform of A000311, define b by 1+sum b_n x^n / n! = exp ( 1 + sum a_n x^n /n!).
E.g.f.: A(x), B(x)=x*A(x) satisfies the differential equation B'(x)=(1+B(x))/(1-B(x)). - Vladimir Kruchinin, Jan 18 2011
From Peter Bala, Sep 05 2011: (Start)
The generating function A(x) satisfies the autonomous differential equation A'(x) = (1+A)/(1-A) with A(0) = 0. Hence the inverse function A^-1(x) = int {t = 0..x} (1-t)/(1+t) = 2*log(1+x)-x, which yields A(x) = -1-2*W(-1/2*exp((x-1)/2)), where W is the Lambert W function.
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx((1+x)/(1-x)*g(x)). Compare with A032188.
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1+t)/(1-t) = 1 + 2*t + 2*t^2 + 2*t^3 + ..., leads to the following combinatorial interpretation for the sequence: a(n) gives the number of plane increasing trees on n vertices where each vertex of outdegree k >=1 can be in one of 2 colors. An example is given below. (End)
A134991 gives (b.+c.)^n = 0^n , for (b_n)=A000311(n+1) and (c_0)=1, (c_1)=-1, and (c_n)=-2* A000311(n) = -A006351(n) otherwise. E.g., umbrally, (b.+c.)^2 = b_2*c_0 + 2 b_1*c_1 + b_0*c_2 =0. - Tom Copeland, Oct 19 2011
G.f.: 1/S(0) where S(k) = 1 - x*(k+1) - x*(k+1)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 18 2011
a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)*sum(j=1..k, (-1)^(j)*C(k,j)* sum(l=0..j, (C(j,l)*(j-l)!*2^(j-l)*(-1)^l*stirling1(n-l+j-1,j-l))/ (n-l+j-1)!)))), n>1, a(1)=1. - Vladimir Kruchinin, Jan 24 2012
E.g.f.: A(x) = exp(B(x))-1 where B(x) is the e.g.f. of A000311. - Vladimir Kruchinin, Sep 25 2012
a(n) = sum_{k=0..n-1} A201637(n-1,k)*2^(n-k-1). - Peter Luschny, Nov 16 2012
G.f.: -1 + 2/Q(0), where Q(k)= 1 - k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ sqrt(2)*n^(n-1)/((2*log(2)-1)^(n-1/2)*exp(n)). - Vaclav Kotesovec, Jul 17 2013
G.f.: Q(0)/(1-x), where Q(k) = 1 - x*(k+1)/( x*(k+1) - (1 -x*(k+1))*(1 -x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020
Conjecture: a(n) = f(0, n-1) where f(n, m) = f(n-1, m) + (n+1)*(f(n, m-1) + f(n+1, m-1)) for n > 0, m > 0 with f(0, m) = f(0, m-1) + f(1, m-1) for m > 0, f(n, 0) = 1 for n >= 0 (see also A338193). - Mikhail Kurkov, Oct 27 2024
EXAMPLE
D^3(1) = (12*x^2+56*x+52)/(x-1)^6. Evaluated at x = 0 this gives a(4) = 52.
a(3) = 8: The 8 possible increasing plane trees on 3 vertices with vertices of outdegree k >= 1 coming in 2 colors, B or W, are
.......................................................
.1B..1B..1W..1W.....1B.......1W........1B........1W....
.|...|...|...|...../.\....../..\....../..\....../..\...
.2B..2W..2B..2W...2...3....2....3....3....2....3....2..
.|...|...|...|.........................................
.3...3...3...3.........................................
G.f. = x + 2*x^2 + 8*x^3 + 52*x^4 + 472*x^5 + 5504*x^6 + 78416*x^7 + ...
MAPLE
read transforms; t1 := 2*ln(1+x)-x; t2 := series(t1, x, 10); t3 := seriestoseries(t2, 'revogf'); t4 := SERIESTOLISTMULT(%);
# N denotes all series-parallel networks, S = series networks, P = parallel networks;
spec := [ N, N=Union(Z, S, P), S=Set(Union(Z, P), card>=2), P=Set(Union(Z, S), card>=2)}, labeled ]: A006351 := n->combstruct[count](spec, size=n);
A006351 := n -> add(combinat[eulerian2](n-1, k)*2^(n-k-1), k=0..n-1):
seq(A006351(n), n=1..18); # Peter Luschny, Nov 16 2012
MATHEMATICA
max = 18; f[x_] := 2*Log[1+x]-x; Rest[ CoefficientList[ InverseSeries[ Series[ f[x], {x, 0, max}], x], x]]*Range[max]! (* Jean-François Alcover, Nov 25 2011 *)
PROG
(Maxima) a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1, n-1)* sum((-1)^(j)*binomial(k, j)*sum((binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l* stirling1(n-l+j-1, j-l))/(n-l+j-1)!, l, 0, j), j, 1, k), k, 1, n-1)); /* Vladimir Kruchinin, Jan 24 2012 */
(Sage) # uses[eulerian2 from A201637]
[A006351(n) for n in (1..18)] # Peter Luschny, Nov 16 2012
(PARI) x='x+O('x^66); Vec(serlaplace(serreverse( 2*log(1+x) - 1*x ))) \\ Joerg Arndt, May 01 2013
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
STATUS
approved