%I #64 Sep 08 2022 08:44:52

%S 0,1,1,1,2,6,19,61,200,670,2286,7918,27770,98424,351983,1268541,

%T 4602752,16799894,61642078,227239086,841230292,3126039364,11656497518,

%U 43601626146,163561902392,615183356156,2319423532024,8764535189296,33187922345210,125912855167740

%N Number of Dyck n-paths starting U^mD^m (an m-pyramid), followed by a pyramid-free Dyck path.

%C Hankel transform is -A128834. - _Paul Barry_, Jul 04 2009

%H Alois P. Heinz, <a href="/A035929/b035929.txt">Table of n, a(n) for n = 0..500</a>

%H J.-L. Baril, S. Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016.

%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.

%H W. Kuszmaul, <a href="http://arxiv.org/abs/1509.08216">Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations</a>, arXiv:1509.08216 [cs.DM], 2015.

%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016.

%F G.f.: A(x) satisfies A^2*(x^2-2*x+2) - A*(x+1) + x = 0.

%F The generating function can be written as x/(1-x) times that of A082989.

%F G.f.: (2*x)/(1+x+(1-x)*sqrt(1-4*x)) = 1/(1-x(1-x)/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - _Paul Barry_, Jul 04 2009

%F From _Gary W. Adamson_, Jul 14 2011: (Start)

%F a(n), n>0; is the upper left term in M^(n-1), where M is the infinite square production matrix:

%F 1, 1, 0, 0, 0, 0, ...

%F 0, 1, 1, 0, 0, 0, ...

%F 1, 1, 1, 1, 0, 0, ...

%F 1, 1, 1, 1, 1, 0, ...

%F 1, 1, 1, 1, 1, 1, ...

%F ... (End)

%F D-finite with recurrence: 2*n*a(n) +4*(-3*n+4)*a(n-1) +(19*n-44)*a(n-2) + (-13*n + 36)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Nov 24 2012

%F a(n) ~ 3 * 4^n / (25 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 12 2014

%F From _Alexander Burstein_, Aug 05 2017: (Start)

%F G.f: A = x/(1-(1-x)*x*C) = x*C/(1+x^2*C^2) = x*C^3/(1+2*x*C^3), where C is the g.f. of A000108.

%F A/x composed with x*C = g.f. of A165543, where A and C are as above. (End)

%e The a(5) = 6 cases are UUUUUDDDDD, UDUUUDUDDD, UDUUUDDUDD, UDUUDUUDDDD, UDUUDUDUDUDD and UUDDUUDUDD.

%p A:= proc(n) option remember; if n=0 then 0 else convert (series ((A(n-1)^2 *(x^2-2*x+2) +x)/ (x+1), x,n+1), polynom) fi end: a:= n-> coeff (A(n), x,n): seq (a(n), n=0..25); # _Alois P. Heinz_, Aug 23 2008

%t CoefficientList[Series[2*x/(1+x+(1-x)*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)

%o (PARI) x='x+O('x^30); concat([0], Vec(2*x/(1+x+(1-x)*sqrt(1-4*x)))) \\ _G. C. Greubel_, Jan 15 2018

%o (Magma) /* Expansion */ Q:=Rationals(); R<x>:=PowerSeriesRing(Q,30); R!(2*x/(1+x+(1-x)*Sqrt(1-4*x))); // _G. C. Greubel_, Jan 15 2018

%Y Cf. A082989.

%K nonn

%O 0,5

%A _N. J. A. Sloane_

%E Edited by _Louis Shapiro_, Feb 16 2005

%E Wrong g.f. removed by _Vaclav Kotesovec_, Feb 12 2014