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
Keywords
Examples
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).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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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);
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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 *)
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