A200757 Noncrossing forests in the regular (n+1)-polygon obtained by a grafting procedure.

1, 3, 13, 68, 395, 2450, 15892, 106489, 731379, 5121392, 36425796, 262425982, 1911063188, 14044679173, 104030937139, 775856119012, 5821085551579, 43906627941144, 332742274685104, 2532358764929916, 19346427410500788, 148312939031577504, 1140578980645677208

Offset

1,2

Comments

The sequence counts noncrossing forests (in the regular (n+1)-polygon) that can be obtained from the three noncrossing forests {0-2}, {0-1-2} and {2-0-1} in the triangle with vertices 0,1,2 by a grafting procedure.

This set describes a suboperad of the WQSYM operad.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..400

F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Formula

G.f. F satisfies: F = x +(x+F)^2/(1-x-F) -F^2/(1-F).

a(n) = sum(i=0..n-1, C(3*n-i-2,2*n-1)*sum(j=0..n, C(j,-2*n+2*j+i)*(-1)^(n-j)*C(n,j)))/n, n>0. - Vladimir Kruchinin, Nov 25 2011

a(n) = Sum_{k=0..n-1} (C(n-1,k)*C(n+2*k,n+k-1))/(n+k). - Vladimir Kruchinin, Mar 02 2013

Recurrence: 2*n*(2*n-1)*(37*n^2 - 157*n + 156)*a(n) = 2*(592*n^4 - 3696*n^3 + 8051*n^2 - 7215*n + 2196)*a(n-1) + 2*(n-3)*(148*n^3 - 702*n^2 + 977*n - 348)*a(n-2) - 5*(n-4)*(n-3)*(37*n^2 - 83*n + 36)*a(n-3). - Vaclav Kotesovec, Aug 15 2013

a(n) ~ c*d^n/n^(3/2), where d = 8.22469154... is the root of the equation 5-8*d-32*d^2+4*d^3=0 and c = 0.11149743370995366254... - Vaclav Kotesovec, Aug 15 2013

Maple

f:= proc(n) option remember; local F;
      if n=0 then 0 else F:= f(n-1);
      convert(series(x+(x+F)^2/(1-x-F)-F^2/(1-F), x, n+1), polynom) fi
    end:
a:= n-> coeff(f(n), x, n):
seq(a(n), n=1..30); # Alois P. Heinz, Nov 22 2011

Mathematica

a[n_] := Sum[ Binomial[3*n - i - 2, 2*n - 1]* Sum[Binomial[j, -2*n + 2*j + i]*(-1)^(n - j)*Binomial[n, j], {j, 0, n}], {i, 0, n - 1}]/n ; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Feb 22 2013, after Vladimir Kruchinin *)

Prog

(Sage)
def suite_ncf(N):
    ano = PowerSeriesRing(QQ, 'x')
    x = ano.gen()
    F = ano.zero().O(1)
    for k in range(N):
        F = x+((x+F)**2/(1-x-F)-F**2/(1-F))
    return F.O(N+1)
(Maxima)
a(n):=sum(binomial(3*n-i-2, 2*n-1)*sum(binomial(j, -2*n+2*j+i)*(-1)^(n-j)*binomial(n, j), j, 0, n), i, 0, n-1)/n; /* Vladimir Kruchinin, Nov 25 2011 */

Crossrefs

Cf. A001764, A054727.

Sequence in context: A042659 A054132 A047149 * A000260 A192737 A125279

Adjacent sequences: A200754 A200755 A200756 * A200758 A200759 A200760

Keyword

nonn

Author

F. Chapoton, Nov 22 2011

Status

approved