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%I #34 Sep 08 2022 08:45:24
%S 1,1,5,16,58,211,781,2920,11006,41746,159154,609324,2341060,9021559,
%T 34855741,134972368,523689718,2035462990,7923732118,30889008112,
%U 120566373676,471134916286,1842964183570,7216096752496,28279240308268,110913181145716,435333520075796,1709861650762900
%N 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).
%C Catalan transform of A034299. - _R. J. Mathar_, Jun 29 2009
%H G. C. Greubel, <a href="/A116914/b116914.txt">Table of n, a(n) for n = 2..1000</a>
%H David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, <a href="http://arxiv.org/abs/1605.06825">Pattern Avoiding Linear Extensions of Rectangular Posets</a>, arXiv:1605.06825 [math.CO], 2016.
%H Colin Defant, <a href="https://arxiv.org/abs/1905.02309">Proofs of Conjectures about Pattern-Avoiding Linear Extensions</a>, arXiv:1905.02309 [math.CO], 2019.
%H E. Deutsch and L. Shapiro, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F a(n) = Sum_{k=0..floor(n/2)} k*A105640(n,k).
%F G.f.: x*(1 + 5*x - (1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x)).
%F a(n+2) = A126258(2*n,n). - _Philippe Deléham_, Mar 13 2007
%F a(n) ~ 2^(2*n-1)/(9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F D-finite with recurrence +2*(-n+1)*a(n) +3*(-n+6)*a(n-1) +3*(13*n-44)*a(n-2) +10*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e 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).
%p 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);
%t 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 *)
%o (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
%o (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
%o (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
%Y Cf. A105640.
%K nonn
%O 2,3
%A _Emeric Deutsch_, May 08 2006
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