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A127540 Number of odd-length branches starting at the root in all ordered trees with n edges. 4

%I

%S 0,1,2,7,21,69,228,773,2659,9275,32715,116511,418377,1513163,5507242,

%T 20155583,74131537,273862373,1015762117,3781095113,14121051487,

%U 52895245133,198681804877,748162728797,2823879525331,10681527145369

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

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

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

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

%F 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 a(n) ~ 3*4^(n+1)/(5*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014

%e 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 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 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 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 (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 Cf. A127538.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Mar 01 2007

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Last modified September 20 22:20 EDT 2019. Contains 327252 sequences. (Running on oeis4.)