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

0, 1, 2, 7, 21, 69, 228, 773, 2659, 9275, 32715, 116511, 418377, 1513163, 5507242, 20155583, 74131537, 273862373, 1015762117, 3781095113, 14121051487, 52895245133, 198681804877, 748162728797, 2823879525331, 10681527145369

Offset: 0

Author

Emeric Deutsch, Mar 01 2007

Keywords

• nonn

Comments

Also number of even-length branches starting at the root in all ordered trees with n+1 edges. - Emeric Deutsch, Mar 03 2007

Also number of Dyck paths of semi-length n+1 with first descent and last ascent of equal size.

Examples

			a(2)=2 because the tree /\ has two odd-length branches starting from the root and the path-tree of length 2 has none.
a(2)=2 because the Dyck paths of semi-length 3 with first descent and last ascent of same size are UUDUDD and UDUDUD.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

Crossrefs

Cf. A127538.

Programs

Maple

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);

Mathematica

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 *)

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

Formula

a(n) = Sum_{k=0..n} k*A127538(n,k).

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.

a(n) ~ 3*4^(n+1)/(5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

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