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%I #55 Jul 26 2022 16:29:13
%S 1,0,1,1,3,5,12,24,55,119,272,612,1411,3247,7565,17667,41561,98099,
%T 232696,553784,1322813,3169065,7614583,18342921,44294991,107200829,
%U 259983346,631718606,1537737567,3749440151,9156561590,22394270034,54845701243,134497468359
%N Number of Motzkin paths of length n with no level steps at even level.
%C Hankel transform of a(n) is A000012. Hankel transform of a(n+1) is 0,-1,0,1,0,-1,0,... or -[x^n](x/(1+x^2)). Hankel transform of a(n+2) is A008619(n+1). - _Paul Barry_, Mar 23 2011
%C Number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U(k) = (k,1) for every positive integer k and down steps D = (1,-1). For instance, for n=5, we have the 5 paths: U(4)D, U(2)U(1)DD, U(1)U(2)DD, U(2)DU(1)D, U(1)DU(2)D. - _José Luis Ramírez Ramírez_, Apr 19 2015
%H Vincenzo Librandi, <a href="/A090345/b090345.txt">Table of n, a(n) for n = 0..1000</a>
%H David Callan, <a href="https://arxiv.org/abs/2112.05241">Some bijections for lattice paths</a>, arXiv:2112.05241 [math.CO], 2021.
%H Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.
%F G.f.: (1 - z - sqrt(1 - 2*z - 3*z^2 + 4*z^3))/(2*z^2).
%F G.f. A(x) satisfies A(x) = A(x/(x-1)). - _Vladeta Jovovic_, Jul 07 2004
%F Also (x*A)^2 = (1-x)*(A-1). - _Vladeta Jovovic_, Jul 07 2004
%F G.f.: 1/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-... (continued fraction). - _Paul Barry_, Apr 08 2009
%F G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x^2/(1-x) (continued fraction); in other words, g.f.: C(x^2/(1-x)) where C(x) is the g.f. for the Catalan numbers (A000108). - _Joerg Arndt_, Mar 18 2011
%F a(0) = 1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*binomial(n-k,k)*A000108(k). - _Paul Barry_, Jul 01 2009
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, n-2k)*A000108(k). - _Paul Barry_, Mar 23 2011
%F The sequence starting with offset 1 = iterates of M*V, leftmost column. M = an infinite tridiagonal matrix with all 1's in the sub and superdiagonals and [0,1,0,1,0,1,0,1,...] as the main diagonal; and the rest zeros. V = vector [1,0,0,0,...]. - _Gary W. Adamson_, Jun 08 2011
%F D-finite with recurrence (n+2)*a(n) + (-2*n-1)*a(n-1) + 3*(-n+1)*a(n-2) + 2*(2*n-5)*a(n-3) = 0. - _R. J. Mathar_, Nov 24 2012
%F a(n) ~ sqrt(34+2*sqrt(17)) * ((1+sqrt(17))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 12 2014
%F a(0) = 1, a(1) = 0; a(n) = a(n-1) + Sum_{k=0..n-2} a(k) * a(n-k-2). - _Ilya Gutkovskiy_, Jul 20 2021
%e a(5)=5 because we have UHDUD, UDUHD, UHUDD, UUDHD and UHHHD, where U=(1,1), D=(1,-1) and H=(1,0).
%t CoefficientList[Series[(1-x-Sqrt[1-2*x-3*x^2+4*x^3])/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%Y Cf. A001006.
%Y First differences of A090344.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Jan 28 2004
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