A114590 Number of peaks at even levels in all hill-free Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1).

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

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(1-4*z))/(2*z^2*(2+z)^2*sqrt(1-4*z)).

a(n) = sum(k*A114588(n+2,k),k=0..n+1).

a(n)=sum{k=0..n, sum{j=0..n-k, C(n-j,k-j)*C(n-j,k)*(j+1)}}; - Paul Barry, Nov 03 2006

Conjecture: 2*(n+2)*a(n) +(-7*n-9)*a(n-1) -18*a(n-2) +2*(-7*n+19)*a(n-3) +4*(-2*n+3)*a(n-4)=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(n-1) + 2*(n+1)*(2*n+1)*(3*n+4)*a(n-2). - 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)) hill-free 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(1-4*z))/2/z^2/(2+z)^2/sqrt(1-4*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[1-4*x])/2/x^2/(2+x)^2/Sqrt[1-4*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Crossrefs

Cf. A114588, A114587, A114515.

Sequence in context: A066796 A104934 A056711 * A133592 A115967 A357641

Adjacent sequences: A114587 A114588 A114589 * A114591 A114592 A114593

Keyword

nonn

Author

Emeric Deutsch, Dec 11 2005

Status

approved