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Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.
3

%I #51 Oct 29 2024 12:22:40

%S 1,2,5,15,52,201,841,3726,17213,82047,400600,1993377,10071777,

%T 51532938,266462229,1390174911,7308741084,38682855225,205940368441,

%U 1102091393574,5925177392573,31987877317887,173337754977904

%N Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

%C Hankel transform is A006215. Invert transform of A155069. - _Michael Somos_, Apr 02 2013

%H Vincenzo Librandi, <a href="/A224071/b224071.txt">Table of n, a(n) for n = 0..200</a>

%H J. Bloom and S. Elizalde, <a href="http://arxiv.org/abs/1211.3442">Pattern avoidance in matchings and partitions</a>, arXiv:1211.3442 [math.CO], 2012; Theorem 6.1.

%H Paul Barry, <a href="http://repository.wit.ie/id/eprint/1379">A study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms</a>, Ph.D Thesis, University College, Cork, Republic of Ireland, 2009.

%H Arnauld Mesinga Mwafise, <a href="https://doi.org/10.31219/osf.io/vg7pr">Computational and Combinatorial Enumeration of Poset Matrices</a>, 2024. See p. 8.

%F G.f.: 4/(3-5*x+sqrt(1-6*x+x^2)).

%F Recurrence: n*a(n) = 9*(n-1)*a(n-1) - 2*(11*n-15)*a(n-2) + 3*(7*n-12)*a(n-3) - 3*(n-3)*a(n-4). - _Vaclav Kotesovec_, May 23 2013

%F a(n) ~ sqrt(884+627*sqrt(2)) * (3+2*sqrt(2))^n / (98*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, May 23 2013

%F 0 = +a(n)*(+9*a(n+1) - 144*a(n+2) + 174*a(n+3) - 81*a(n+4) + 12*a(n+5)) + a(n+1)*(+18*a(n+1) + 399*a(n+2) - 597*a(n+3) + 318*a(n+4) - 57*a(n+5)) + a(n+2)*(-300*a(n+2) + 538*a(n+3) - 255*a(n+4) + 52*a(n+5)) + a(n+3)*(-126*a(n+3) + 73*a(n+4) - 18*a(n+5)) + a(n+4)*(+a(n+5)) if n>=0. - _Michael Somos_, Mar 28 2014

%F a(n) = Sum_{k=0..n}((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*Sum_{i=0..n-k}(binomial(n+1,n-k-i)*binomial(n+i,n)))/(2*(n+1)). - _Vladimir Kruchinin_, Mar 08 2016

%e a(2) = 5 because we have HH, UDH, HUD, UDUD and UUDD.

%e G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 52*x^4 + 201*x^5 + 841*x^6 + ...

%t CoefficientList[Series[4/(3-5*x+Sqrt[x^2-6*x+1]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 23 2013 *)

%t a[ n_] := SeriesCoefficient[ (3 - 5 x - Sqrt[ 1 - 6 x + x^2]) / (2 - 6 x + 6 x^2), {x, 0, n}]; (* _Michael Somos_, Mar 28 2014 *)

%o (PARI) z='z+O('z^66); Vec(4/(3-5*z+sqrt(1-6*z+z^2))) /* _Joerg Arndt_, Mar 30 2013 */

%o (Maxima)

%o a(n):=sum((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*sum(binomial(n+1,n-k-i)*binomial(n+i,n),i,0,n-k),k,0,n)/(2*(n+1)); /* _Vladimir Kruchinin_, Mar 08 2016*/

%Y Cf. A001003, A007564, A010892, A059231.

%K nonn

%O 0,2

%A _José Luis Ramírez Ramírez_, Mar 30 2013