Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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