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
KEYWORD
nonn
AUTHOR
F. Chapoton, Nov 22 2011
STATUS
approved