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A200757 Noncrossing forests in the regular (n+1)-polygon obtained by a grafting procedure. 1
1, 3, 13, 68, 395, 2450, 15892, 106489, 731379, 5121392, 36425796, 262425982, 1911063188, 14044679173, 104030937139, 775856119012, 5821085551579, 43906627941144, 332742274685104, 2532358764929916, 19346427410500788, 148312939031577504, 1140578980645677208 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 22 2013, after Vladimir Kruchinin *)

PROG

(Sage)

def suite_ncf(N):

      ano=PowerSeriesRing(QQ, 'x')

    x=ano.gen()

    F=ano(0).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

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Last modified May 20 02:50 EDT 2019. Contains 323412 sequences. (Running on oeis4.)