cp's OEIS Frontend

A062236 Sum of the levels of all nodes in all noncrossing trees with n edges.

%I A062236 #51 Jul 29 2025 08:36:59
%S A062236 1,8,58,408,2831,19496,133638,913200,6226591,42387168,288194424,
%T A062236 1957583712,13286865060,90126841064,611029568078,4140789069408,
%U A062236 28050809681679,189964288098632,1286119453570746,8705397371980728,58912358137385559,398607288093924192,2696583955707785256
%N A062236 Sum of the levels of all nodes in all noncrossing trees with n edges.
%H A062236 Harry J. Smith, <a href="/A062236/b062236.txt">Table of n, a(n) for n=1..200</a>
%H A062236 Emeric Deutsch and M. Noy, <a href="http://dx.doi.org/10.1007/978-3-662-04166-6_65">New statistics on non-crossing trees</a>, in: Formal Power Series and Algebraic Combinatorics (Proceedings of the 12th International Conference, FPSAC'00, Moscow, Russia, 2000), pp. 667-676, Springer, Berlin, 2000.
%H A062236 Emeric Deutsch and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00366-1">Statistics on non-crossing trees</a>, Discrete Math., 254 (2002), 75-87 (see Th. 6). [From _N. J. A. Sloane_, Dec 17 2012]
%H A062236 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%H A062236 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F A062236 G.f.: g*(g-1)/(3-2*g)^2, where function g=g(x) satisfies g=1+xg^3, and can be expressed as g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x). [Corrected by _Max Alekseyev_, Oct 27 2022]
%F A062236 g(x) = Sum_{n >= 0} binomial(3*n,n) / (2*n+1) * x^n. - _Max Alekseyev_, Oct 27 2022
%F A062236 Recurrence: 8*n*(2*n-1)*a(n) = 6*(36*n^2-45*n+10)*a(n-1) - 81*(3*n-5)*(3*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F A062236 a(n) ~ 3^(3*n)/2^(2*n+2). - _Vaclav Kotesovec_, Oct 13 2012
%F A062236 a(n) = Sum_{i=0..n-1} C(3*i-1,i)*C(3*(n-i),n-i-1). - _Vladimir Kruchinin_, Jun 09 2020
%F A062236 a(n) = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1-n, -n+4/3], [-n, -n+1/3], -1/2). The a(n) are values of the polynomials A358091. - _Peter Luschny_, Oct 28 2022
%F A062236 From _Seiichi Manyama_, Jul 26 2025: (Start)
%F A062236 G.f.: g/(1-3*g)^2 where g*(1-g)^2 = x.
%F A062236 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/2) * log( Sum_{k>=0} binomial(3*k-1,k)*x^k ). (End)
%F A062236 From _Seiichi Manyama_, Jul 29 2025: (Start)
%F A062236 a(n) = Sum_{k=0..n-1} binomial(3*k-1+l,k) * binomial(3*n-3*k-l,n-k-1) for every real number l. This is a generalization of a formula by Vladimir Kruchinin, Jun 09 2020.
%F A062236 a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(3*n,k).
%F A062236 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(2*n+k,k). (End)
%p A062236 a := n -> add(2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n,i),i=0..n-1)/n;
%p A062236 A062236 := n -> 2^(n-2)*(3*n-1)*hypergeom([-3*n,1-n,-n+4/3], [-n,-n+1/3], -1/2):
%p A062236 seq(simplify(A062236(n)), n = 1..29); # _Peter Luschny_, Oct 28 2022
%t A062236 Table[Sum[2^(n-2-k)*(n-k)*(3*n-3*k-1)*Binomial[3*n,k],{k,0,n-1}]/n,{n,1,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)
%o A062236 (PARI) { for (n=1, 200, a=sum(i=0, n-1, 2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n, i))/n; write("b062236.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 03 2009
%Y A062236 Cf. A001764, A036829, A358091.
%Y A062236 Cf. A006256, A036829, A075045, A386617.
%K A062236 nonn
%O A062236 1,2
%A A062236 _Emeric Deutsch_, Jun 30 2001