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 A109196 Number of returns to the x-axis from above (i.e., d steps hitting the x-axis) in all Grand Motzkin paths of length n. 4
 1, 3, 11, 35, 112, 350, 1087, 3351, 10286, 31460, 95966, 292110, 887629, 2693423, 8163367, 24717575, 74778718, 226066940, 683006416, 2062412936, 6224697139, 18779180645, 56633215930, 170733734210, 514559844007, 1550364293145 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS A Grand Motzkin path of length n is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0). The substitution x->x/(1+x+x^2), the inverse Motzkin transform, yields a g.f. for the sequence 0,0,2,2,6,4,..., that is 0 followed by 2*A026741(n-1). - R. J. Mathar, Nov 10 2008 LINKS G. C. Greubel, Table of n, a(n) for n = 2..1000 FORMULA G.f.: (1-z-sqrt(1-2*z-3*z^2)) / (2*(1-2*z-3*z^2)). a(n) = Sum_{k=0..floor(n/2)} k*A109195(n,k). a(n) = (1/2) * A109194(n). From Benedict W. J. Irwin, Nov 02 2016: (Start) Conjecture: a(n) = (2*(-1)^n + 2*3^n + (2^n*(2*n - 1)!!*(3*A - 4*B))/n! - 3*(n + 1)*C)/8. A = 2F1(1-n,-n; 1/2-n; 1/4). B = 2F1(-n,-n; 1/2-n; 1/4). 2^n*(2*n - 1)!!*(3*A - 4*B))/n! = A103872(n-2). C = 3F2(1-n,(1-n)/2,-n/2; 2,-n-1; 4) = A025565(n)/n. (End) a(n) ~ 3^n/4 * (1-sqrt(3/(Pi*n))). - Vaclav Kotesovec, Nov 05 2016 D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(-2*n+3)*a(n-2) +3*(4*n-9)*a(n-3) +9*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 08 2021 EXAMPLE a(3)=3 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hu(d), hdu, u(d)h, duh, uh(d) and dhu; they have a total of 3 returns from above to the x-axis (shown between parentheses). MAPLE g:=(1-z-sqrt(1-2*z-3*z^2))/2/(1-2*z-3*z^2): gser:=series(g, z=0, 32): seq(coeff(gser, z^n), n=2..30); MATHEMATICA Rest[Rest[CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 3 x^2]) / (2 (1 - 2 x - 3 x^2)), {x, 0, 35}], x]]] (* Vincenzo Librandi, Nov 04 2016 *) CROSSREFS Cf. A109194, A109195. Sequence in context: A025181 A004054 A068995 * A032637 A034576 A125672 Adjacent sequences: A109193 A109194 A109195 * A109197 A109198 A109199 KEYWORD nonn AUTHOR Emeric Deutsch, Jun 22 2005 STATUS approved

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Last modified February 3 14:36 EST 2023. Contains 360035 sequences. (Running on oeis4.)