cp's OEIS Frontend

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

%I A200757 #38 Feb 29 2020 18:02:18
%S A200757 1,3,13,68,395,2450,15892,106489,731379,5121392,36425796,262425982,
%T A200757 1911063188,14044679173,104030937139,775856119012,5821085551579,
%U A200757 43906627941144,332742274685104,2532358764929916,19346427410500788,148312939031577504,1140578980645677208
%N A200757 Noncrossing forests in the regular (n+1)-polygon obtained by a grafting procedure.
%C A200757 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.
%C A200757 This set describes a suboperad of the WQSYM operad.
%H A200757 Alois P. Heinz, <a href="/A200757/b200757.txt">Table of n, a(n) for n = 1..400</a>
%H A200757 F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%F A200757 G.f. F satisfies: F = x +(x+F)^2/(1-x-F) -F^2/(1-F).
%F A200757 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
%F A200757 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
%F A200757 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
%F A200757 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
%p A200757 f:= proc(n) option remember; local F;
%p A200757       if n=0 then 0 else F:= f(n-1);
%p A200757       convert(series(x+(x+F)^2/(1-x-F)-F^2/(1-F), x, n+1), polynom) fi
%p A200757     end:
%p A200757 a:= n-> coeff(f(n), x, n):
%p A200757 seq(a(n), n=1..30); # _Alois P. Heinz_, Nov 22 2011
%t A200757 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_ *)
%o A200757 (Sage)
%o A200757 def suite_ncf(N):
%o A200757     ano = PowerSeriesRing(QQ,'x')
%o A200757     x = ano.gen()
%o A200757     F = ano.zero().O(1)
%o A200757     for k in range(N):
%o A200757         F = x+((x+F)**2/(1-x-F)-F**2/(1-F))
%o A200757     return F.O(N+1)
%o A200757 (Maxima)
%o A200757 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 */
%Y A200757 Cf. A001764, A054727.
%K A200757 nonn
%O A200757 1,2
%A A200757 _F. Chapoton_, Nov 22 2011