cp's OEIS Frontend

A127540 Number of odd-length branches starting at the root in all ordered trees with n edges.

%I A127540 #22 Jul 26 2022 15:16:39
%S A127540 0,1,2,7,21,69,228,773,2659,9275,32715,116511,418377,1513163,5507242,
%T A127540 20155583,74131537,273862373,1015762117,3781095113,14121051487,
%U A127540 52895245133,198681804877,748162728797,2823879525331,10681527145369
%N A127540 Number of odd-length branches starting at the root in all ordered trees with n edges.
%C A127540 Also number of even-length branches starting at the root in all ordered trees with n+1 edges. - _Emeric Deutsch_, Mar 03 2007
%C A127540 Also number of Dyck paths of semi-length n+1 with first descent and last ascent of equal size.
%H A127540 G. C. Greubel, <a href="/A127540/b127540.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%F A127540 a(n) = Sum_{k=0..n} k*A127538(n,k).
%F A127540 G.f.: x*C/(1-x^2-x*C-x^2*C), where C = (1-sqrt(1-4*x))/(2*x) is the Catalan function.
%F A127540 a(n) ~ 3*4^(n+1)/(5*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F A127540 D-finite with recurrence (n+2)*a(n) -4*n*a(n-1) +(-n-8)*a(n-2) +2*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e A127540 a(2)=2 because the tree /\ has two odd-length branches starting from the root and the path-tree of length 2 has none.
%e A127540 a(2)=2 because the Dyck paths of semi-length 3 with first descent and last ascent of same size are UUDUDD and UDUDUD.
%p A127540 C:=(1-sqrt(1-4*z))/2/z: g:=z*C/(1-z^2-z*C-z^2*C): gser:=series(g,z=0,32): seq(coeff(gser,z,n),n=0..29);
%t A127540 CoefficientList[Series[x (1 - (1 - 4*x)^(1/2))/(2*x)/(1 - x^2 - x *(1 - (1 - 4*x)^(1/2)) /(2*x) - x^2*(1 - (1 - 4*x)^(1/2))/(2*x)), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o A127540 (PARI) concat([0], Vec(x*(1 - (1 - 4*x)^(1/2))/(2*x)/(1 - x^2 - x*(1 - (1 - 4*x)^(1/2)) /(2*x) - x^2*(1 - (1 - 4*x)^(1/2))/(2*x)) + O(x^50))) \\ _G. C. Greubel_, Jan 31 2017
%Y A127540 Cf. A127538.
%K A127540 nonn
%O A127540 0,3
%A A127540 _Emeric Deutsch_, Mar 01 2007