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A105641 Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1). 1
0, 1, 2, 5, 14, 39, 111, 322, 947, 2818, 8470, 25677, 78420, 241061, 745265, 2315794, 7228702, 22656505, 71273364, 224965675, 712249471, 2261326010, 7197988973, 22966210236, 73437955105, 235307698544, 755395560220, 2429293941019 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
COMMENTS
a(n) = A105640(n,0).
LINKS
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
FORMULA
G.f.: [(1+z)^2-sqrt((1+z^2)^2-4z)]/[2z(2+z+z^2)]-1.
D-finite with recurrence 2*(n+1)*a(n) +(-7*n+5)*a(n-1) +(n-5)*a(n-2) +2*(-n-1)*a(n-3) +2*(2*n-7)*a(n-4) +(n-5)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
EXAMPLE
a(4)=2 because we have UUDUDUDD and UUUDUDDD.
MAPLE
G:=((1+z)^2-sqrt((1+z^2)^2-4*z))/2/z/(2+z+z^2)-1: Gser:=series(G, z=0, 36): seq(coeff(Gser, z^n), n=2..32);
CROSSREFS
Cf. A118995.
Sequence in context: A026135 A201778 A367655 * A027035 A102406 A307754
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 08 2006
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)