A005750 Number of planted matched trees with n nodes. (Formerly M2855)

1, 1, 3, 10, 39, 160, 702, 3177, 14830, 70678, 342860, 1686486, 8393681, 42187148, 213828802, 1091711076, 5609297942, 28982708389, 150496728594, 784952565145, 4110491658233, 21602884608167, 113907912618599, 602414753753310, 3194684310627727, 16984594260224529

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

Index entries for sequences related to rooted trees

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

Cf. A000151, A058870, A058866, A054581, A245870.

Sequence in context: A050385 A296195 A123768 * A151068 A151069 A151070

Adjacent sequences: A005747 A005748 A005749 * A005751 A005752 A005753

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms, formula and comment from Christian G. Bower, Dec 15 1999

Status

approved