

A114590


Number of peaks at even levels in all hillfree Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1).


4



1, 2, 8, 28, 103, 382, 1432, 5408, 20546, 78436, 300636, 1156188, 4459267, 17241526, 66807856, 259361920, 1008598126, 3928120924, 15319329472, 59817190552, 233826979750, 914962032172, 3583556424208, 14047386554368, 55108441878868
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OFFSET

0,2


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000


FORMULA

G.f.: (1+2*z^2(1+2*z)*sqrt(14*z))/(2*z^2*(2+z)^2*sqrt(14*z)).
a(n) = sum(k*A114588(n+2,k),k=0..n+1).
a(n)=sum{k=0..n, sum{j=0..nk, C(nj,kj)*C(nj,k)*(j+1)}};  Paul Barry, Nov 03 2006
Conjecture: 2*(n+2)*a(n) +(7*n9)*a(n1) 18*a(n2) +2*(7*n+19)*a(n3) +4*(2*n+3)*a(n4)=0.  R. J. Mathar, Nov 15 2012
Recurrence: 2*n*(n+2)*(3*n+1)*a(n) = (21*n^3 + 34*n^2 + n  8)*a(n1) + 2*(n+1)*(2*n+1)*(3*n+4)*a(n2).  Vaclav Kotesovec, Feb 12 2014
a(n) ~ 4^(n+2) / (9*sqrt(Pi*n)).  Vaclav Kotesovec, Feb 12 2014


EXAMPLE

a(1)=2 because in the 2 (=A000957(4)) hillfree Dyck paths of semilength 3, namely UUUDDD and U(UD)(UD)D (U=(1,1), D=(1,1)) we have altogether 2 peaks at even level (shown between parentheses).


MAPLE

G:=(1+2*z^2(1+2*z)*sqrt(14*z))/2/z^2/(2+z)^2/sqrt(14*z): Gser:=series(G, z=0, 30): 1, seq(coeff(Gser, z^n), n=1..25);


MATHEMATICA

CoefficientList[Series[(1+2*x^2(1+2*x)*Sqrt[14*x])/2/x^2/(2+x)^2/Sqrt[14*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)


CROSSREFS

Cf. A114588, A114587, A114515.
Sequence in context: A066796 A104934 A056711 * A133592 A115967 A150714
Adjacent sequences: A114587 A114588 A114589 * A114591 A114592 A114593


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 11 2005


STATUS

approved



