

A127540


Number of oddlength branches starting at the root in all ordered trees with n edges.


4



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
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OFFSET

0,3


COMMENTS

Also number of evenlength branches starting at the root in all ordered trees with n+1 edges.  Emeric Deutsch, Mar 03 2007
Also number of Dyck paths of semilength n+1 with first descent and last ascent of equal size.


LINKS

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


FORMULA

a(n) = Sum_{k=0..n} k*A127538(n,k).
G.f.: x*C/(1x^2x*Cx^2*C), where C = (1sqrt(14*x))/(2*x) is the Catalan function.
a(n) ~ 3*4^(n+1)/(5*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 21 2014


EXAMPLE

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


MAPLE

C:=(1sqrt(14*z))/2/z: g:=z*C/(1z^2z*Cz^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 *)


PROG

(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


CROSSREFS

Cf. A127538.
Sequence in context: A274203 A330058 A220726 * A319852 A060900 A305850
Adjacent sequences: A127537 A127538 A127539 * A127541 A127542 A127543


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 01 2007


STATUS

approved



