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 A090345 Number of Motzkin paths of length n with no level steps at even level. 9
 1, 0, 1, 1, 3, 5, 12, 24, 55, 119, 272, 612, 1411, 3247, 7565, 17667, 41561, 98099, 232696, 553784, 1322813, 3169065, 7614583, 18342921, 44294991, 107200829, 259983346, 631718606, 1537737567, 3749440151, 9156561590, 22394270034, 54845701243, 134497468359 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 David Callan, Some bijections for lattice paths, arXiv:2112.05241 [math.CO], 2021. Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562. FORMULA G.f.: (1 - z - sqrt(1 - 2*z - 3*z^2 + 4*z^3))/(2*z^2). G.f. A(x) satisfies A(x) = A(x/(x-1)). - Vladeta Jovovic, Jul 07 2004 Also (x*A)^2 = (1-x)*(A-1). - Vladeta Jovovic, Jul 07 2004 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 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 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 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, n-2k)*A000108(k). - Paul Barry, Mar 23 2011 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 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 a(n) ~ sqrt(34+2*sqrt(17)) * ((1+sqrt(17))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014 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 EXAMPLE a(5)=5 because we have UHDUD, UDUHD, UHUDD, UUDHD and UHHHD, where U=(1,1), D=(1,-1) and H=(1,0). MATHEMATICA 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 *) CROSSREFS Cf. A001006. First differences of A090344. Sequence in context: A026786 A027246 A185087 * A186334 A303587 A151524 Adjacent sequences: A090342 A090343 A090344 * A090346 A090347 A090348 KEYWORD nonn AUTHOR Emeric Deutsch, Jan 28 2004 STATUS approved

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Last modified March 31 20:29 EDT 2023. Contains 361672 sequences. (Running on oeis4.)