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%I #24 Jul 26 2022 14:08:49
%S 1,0,2,5,16,51,168,565,1934,6716,23604,83806,300154,1083137,3934404,
%T 14374413,52787766,194746632,721435884,2682522918,10008240456,
%U 37455101382,140569122624,528926230530,1994980278636,7541234323096
%N Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).
%C Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).
%H G. C. Greubel, <a href="/A118973/b118973.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%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 G.f.: (1-x)*C*F, where F = (1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)) and C = (1-sqrt(1-4*x))/(2*x) is the Catalan function.
%F a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F a(n) = (Sum_{k=0..n-1}((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1))))+(-1)^(n). - _Vladimir Kruchinin_. Mar 06 2016
%F D-finite with recurrence +2*(n+2)*a(n) +(-7*n-2)*a(n-1) +2*(-3*n+1)*a(n-2) +(7*n-26)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, where u=(1,1),d=(1,-1) (the first descents are shown between parentheses).
%p F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=(1-z)*C*F: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..28);
%p A118973List := proc(m) local A, P, n; A := [1,0]; P := [1,0];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
%p A := [op(A), P[-1]] od; A end: A118973List(26); # _Peter Luschny_, Mar 26 2022
%t CoefficientList[Series[(1-x)*(1-Sqrt[1-4*x])/x/(3-Sqrt[1-4*x])*(1-Sqrt[1-4*x])/2/x, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (Maxima)
%o a(n):=(sum((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1)),k,0,n-1))+(-1)^(n); /* _Vladimir Kruchinin_. Mar 06 2016 */
%o (PARI) x='x+O('x^25); Vec((1-x)*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1-sqrt(1-4*x))/2/x) \\ _G. C. Greubel_, Feb 08 2017
%Y Cf. A000108, A000957, A118972, A001558.
%K nonn
%O 0,3
%A _Emeric Deutsch_, May 08 2006
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