A007858 G.f. is 1 - 1/f(x), where f(x) = 1+x+3*x^2+9*x^3+32*x^4+... is 1/x times g.f. for A063020.

1, 2, 4, 13, 44, 164, 636, 2559, 10556, 44440, 190112, 824135, 3612244, 15981632, 71277736, 320121747, 1446537564, 6571858168, 30000766128, 137544893940, 633051803120, 2923867281660, 13547594977500, 62955434735505, 293336372858724, 1370149533359784, 6414423856436816

Offset

1,2

Comments

Number of maximal independent sets in rooted plane trees on n nodes. - Olivier Gérard, Jul 05 2001

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

A. Mironov and A. Morozov, Algebra of quantum C-polynomials, arXiv:2009.11641 [hep-th], 2020.

Index entries for sequences related to rooted trees

Formula

a(n+1) = Sum_{k = 1..n} ( binomial(n+k,k)/(n+k)*Sum_{j = 0..k} ( binomial(j,n-k-j+1)*binomial(k,j)*(-1)^(n+k-j+1) ) ) + C(n), where C(n) is a Catalan number. - Vladimir Kruchinin, Nov 13 2014

Recurrence: 16*(n-1)*n*(2*n-3)*(17*n^2 - 81*n + 96)*a(n) = (n-1)*(1819*n^4 - 14124*n^3 + 40377*n^2 - 50320*n + 23040)*a(n-1) + 8*(2*n-5)*(4*n-11)*(4*n-9)*(17*n^2 - 47*n + 32)*a(n-2). - Vaclav Kotesovec, Nov 14 2014

Asymptotics (Klazar, 1997): a(n) ~ sqrt(5731-4635/sqrt(17)) * ((107+51*sqrt(17))/64)^n / (256 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 14 2014

Maple

series(1-x/RootOf(_Z-_Z^2-_Z^3+_Z^4-x), x=0, 20); # Mark van Hoeij, May 28 2013

Mathematica

Rest[CoefficientList[1-x/InverseSeries[Series[x-x^2-x^3+x^4, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[Sum[Binomial[n + k, k]/(n + k)*Sum[(Binomial[j, n - k - j + 1]*
Binomial[k, j]*(-1)^(n + k - j + 1)), {j, 0, k}], {k, 1, n}] + CatalanNumber[n], {n, 0, 50}] (* G. C. Greubel, Feb 15 2017 *)

Prog

(PARI) x='x+O('x^66); Vec(1-x/serreverse(x-x^2-x^3+x^4)) \\ Joerg Arndt, May 28 2013
(Maxima)
a(n):=sum(binomial(n+k, k)/(n+k)*sum(binomial(j, n-k-j+1)*binomial(k, j)*(-1)^(n+k-j+1), j, 0, k), k, 1, n)+1/(n+1)*binomial(2*n, n); // Vladimir Kruchinin, Nov 13 2014

Crossrefs

Cf. A000108.

Sequence in context: A001548 A193057 A115600 * A337797 A300931 A286074

Adjacent sequences: A007855 A007856 A007857 * A007859 A007860 A007861

Keyword

nonn

Author

Martin Klazar, Mar 15 1996

Status

approved