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 A217312 Number of Motzkin paths of length n with no level steps at height 1. 6
 1, 1, 2, 3, 6, 11, 23, 48, 107, 244, 578, 1402, 3485, 8826, 22729, 59340, 156766, 418319, 1125956, 3053400, 8334578, 22881070, 63135802, 175000959, 487042069, 1360440914, 3812681435, 10717405374, 30209571942, 85368323429, 241801775480, 686366436772 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 2*(1+x)/(2-x-3*x^2+x*sqrt(1-2*x-3*x^2)) = 1/(1-x-x^2*R), where R is the g.f. of Riordan numbers (A005043). a(n) = 1+Sum_{k=0..(n-1)/2}((k+1)*Sum_{i=0..n-2*k-1}(((Sum_{j=0..i}((-1)^(j-i)*binomial(k+i+1,i-j)*binomial(k+2*j,j)))*binomial(n-k-i-1,k+1))/(k+i+1))). - Vladimir Kruchinin, Mar 12 2016 D-finite with recurrence (-n+1)*a(n) +(4*n-7)*a(n-1) -3*a(n-2) +(-11*n+32)*a(n-3) +3*(n-1)*a(n-4) +9*(n-4)*a(n-5)=0. - R. J. Mathar, Sep 24 2016 a(n) ~ 3^(n - 1/2) / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 20 2019 EXAMPLE The a(4) = 6 paths are HHHH, UDUD, HUDH, UDHH, HHUD, UUDD. MAPLE b:= proc(n, y) option remember;        `if`(y>n, 0, `if`(n=0, 1, `if`(y<>1, b(n-1, y), 0)+        `if`(y>0, b(n-1, y-1), 0)+ b(n-1, y+1)))     end: a:= n-> b(n, 0): seq(a(n), n=0..40);  # Alois P. Heinz, Mar 18 2013 MATHEMATICA b[n_, y_] := b[n, y] = If[y>n, 0, If[n == 0, 1, If[y != 1, b[n-1, y], 0] + If[y>0, b[n-1, y-1], 0] + b[n-1, y+1]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 22 2017, after Alois P. Heinz *) PROG (Maxima) a(n):=(sum((k+1)*sum(((sum((-1)^(j-i)*binomial(k+i+1, i-j)*binomial(k+2*j, j), j, 0, i))*binomial(n-k-i-1, k+1))/(k+i+1), i, 0, n-2*k-1), k, 0, (n-1)/2))+1; /* Vladimir Kruchinin, Mar 12 2016 */ CROSSREFS Cf. A001006, A005043. Sequence in context: A277796 A123465 A000055 * A006787 A176425 A000992 Adjacent sequences:  A217309 A217310 A217311 * A217313 A217314 A217315 KEYWORD nonn AUTHOR José Luis Ramírez Ramírez, Mar 17 2013 STATUS approved

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Last modified October 1 11:36 EDT 2020. Contains 337443 sequences. (Running on oeis4.)