OFFSET
1,3
COMMENTS
When convolved with itself gives A000151.
Number of rooted trees with n nodes and edges not attached to root are 2-colored or oriented.
Also number of 2-trees (with 2n+1 cells) rooted at a symmetric end-edge. - Vladeta Jovovic, Aug 22 2001
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.5.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..1325 (terms 1..500 from Alois P. Heinz)
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. See page 20, line -3. - From N. J. A. Sloane, Dec 15 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 428
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
N. J. A. Sloane, Transforms
FORMULA
a(n+1) is Euler transform of A000151.
G.f.: A(x) = x*exp( A(x)^2/x + A(x^2)^2/(2x^2) + A(x^3)^2/(3x^3) + ... + A(x^n)^2/(n*x^n) + ...). - Paul D. Hanna
G.f.: sqrt(B(x)/x) where B(x) is the g.f. of A000151. - Andrew Howroyd, May 13 2018
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232..., c = 0.06185402386554883780092844840921448929211072031752507960399709674242810089... - Vaclav Kotesovec, Sep 12 2014, updated Dec 26 2020
EXAMPLE
A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 160*x^6 + 702*x^7 + ...
MAPLE
A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(add((A(n-1)(x^k))^2/(k*x^k), k=1..2*n)), x=0, 2*n), polynom), x) fi end: a:= n-> coeff(series(A(n)(x), x=0, n+1), x, n): seq(a(n), n=1..23); # Alois P. Heinz, Aug 20 2008
MATHEMATICA
max = 23; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[ Log[f[x]/x] - Sum[f[x^k]^2/(k*x^k), {k, 1, max}], {x, 0, max}], x]; eqns = Rest[ Thread[coes == 0]]; s[2] = Solve[eqns[[1]], c[2]][[1]]; Do[eqns = Rest[eqns] /. s[k-1]; s[k] = Solve[ eqns[[1]], c[k]][[1]], {k, 3, max}]; Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 2, max}]] (* Jean-François Alcover, Oct 25 2011, after g.f. *)
terms = 26; (* B = g.f. of A000151 *) B[_] = 0; Do[B[x_] = x*Exp[2*Sum[ B[x^k]/k, {k, 1, terms}]] + O[x]^terms // Normal, terms];
A[x_] = Exp[Sum[B[x^k]/k, {k, 1, terms}]] + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)
PROG
(PARI) seq(N) = {my(A=vector(N, j, 1)); for(n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); Vec(sqrt(Ser(A)))} \\ Andrew Howroyd, May 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms, formula and comment from Christian G. Bower, Dec 15 1999
STATUS
approved