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%I #36 Mar 07 2023 11:19:56
%S 1,1,2,4,9,22,56,146,388,1048,2869,7942,22192,62510,177308,506008,
%T 1451866,4185788,12119696,35227748,102753800,300672368,882373261,
%U 2596389190,7658677856,22642421206,67081765932,199128719896,592179010350,1764044315540,5263275015120
%N Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 3).
%C The antidiagonal sums of A091894 equal this sequence. - _Johannes W. Meijer_, Sep 13 2012
%H Robert Israel, <a href="/A152225/b152225.txt">Table of n, a(n) for n = 0..2025</a>
%H Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, <a href="http://jl.baril.u-bourgogne.fr/forest.pdf">Forests and pattern-avoiding permutations modulo pure descents</a>, Permutation Patterns 2017, Reykjavik University, Iceland, June 26-30, 2017. See Section 5.
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%H Shu-Chung Liu, Jun Ma, and Yeong-Nan Yeh, <a href="http://dx.doi.org/10.1111/j.1467-9590.2008.00415.x">Dyck Paths with Peak- and Valley-Avoiding Sets</a>, Stud. Appl Math. 121 (3) (2008) 263-289.
%F G.f.: (1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3))/(2*x^2).
%F Conjecture: (n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0. - _R. J. Mathar_, Aug 14 2012
%F This conjecture follows from the differential equation (4*x^4-4*x^3+4*x^2-x)*y' + (2*x^3-4*x^2+6*x-2)*y - 2*x^3+2*x^2-3*x+2=0 satisfied by the g.f. - _Robert Israel_, Jan 09 2018
%p f:= gfun:-rectoproc({(n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0,a(0)=1,a(1)=1,a(2)=2,a(3)=4},a(n),remember):
%p map(f, [$0..40]); # _Robert Israel_, Jan 09 2018
%t CoefficientList[Series[(1 - 2 x + 2 x^2 - Sqrt[1 - 4 x + 4 x^2 - 4 x^3])/(2 x^2), {x, 0, 30}], x] (* _Michael De Vlieger_, Jan 09 2018 *)
%Y Cf. A091561, A025265, A025247. - _R. J. Mathar_, Dec 03 2008
%K nonn
%O 0,3
%A Majun (majun(AT)math.sinica.edu.tw), Nov 29 2008
%E Edited by _Emeric Deutsch_, Dec 20 2008
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