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A116914 Number of UUDD's, where U=(1,1) and D=(1,-1), in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1). 3
1, 1, 5, 16, 58, 211, 781, 2920, 11006, 41746, 159154, 609324, 2341060, 9021559, 34855741, 134972368, 523689718, 2035462990, 7923732118, 30889008112, 120566373676, 471134916286, 1842964183570, 7216096752496, 28279240308268, 110913181145716, 435333520075796, 1709861650762900 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

Catalan transform of A034299. - R. J. Mathar, Jun 29 2009

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..1000

David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.

Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

FORMULA

a(n) = Sum_{k=0..floor(n/2)} k*A105640(n,k).

G.f.: x*(1 + 5*x - (1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x)).

a(n+2) = A126258(2*n,n). - Philippe Deléham, Mar 13 2007

a(n) ~ 2^(2*n-1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's (shown between parentheses).

MAPLE

G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G, z=0, 31): seq(coeff(Gser, z^n), n=2..28);

MATHEMATICA

Rest[Rest[CoefficientList[Series[x*(1+5*x-(1-x)*Sqrt[1-4*x])/2/(2+x)^2/Sqrt[1-4*x], {x, 0, 40}], x]]] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(PARI) my(x='x+O('x^40)); Vec(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2 *sqrt(1-4*x))) \\ G. C. Greubel, Feb 08 2017

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( x*(1 + 5*x-(1-x)*Sqrt(1-4*x))/(2*(2+x)^2*Sqrt(1-4*x)) )); // G. C. Greubel, May 08 2019

(Sage) a=(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x))).series(x, 40).coefficients(x, sparse=False); a[2:] # G. C. Greubel, May 08 2019

CROSSREFS

Cf. A105640.

Sequence in context: A226973 A006217 A281870 * A047103 A226897 A077235

Adjacent sequences:  A116911 A116912 A116913 * A116915 A116916 A116917

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 08 2006

STATUS

approved

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Last modified December 3 16:20 EST 2020. Contains 338906 sequences. (Running on oeis4.)