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 A108436 Number of returns to the x-axis in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1). 1
 2, 14, 106, 862, 7378, 65550, 599002, 5594942, 53181730, 512784142, 5003410762, 49312114334, 490192537586, 4909102791694, 49482525122490, 501626536004734, 5111038278845506, 52312236295906830, 537605889306476074 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Table of n, a(n) for n=1..19. Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370. FORMULA a(n) = Sum_{k=1..n} k*A108435(k). a(n) = A032349(n+1) - A027307(n). a(n) = 2 + (1/n)*Sum_{j=0..n-2} (3n-j)*2^(n-j)*binomial(n, j)*binomial(2n, n-j-1)/(n+j+2). G.f.: A^2-A, where A=1+zA^2+zA^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307). D-finite with recurrence 3*(n+1)*(2*n+1)*a(n) +3*(-34*n^2+18*n-1)*a(n-1) +(394*n^2-1197*n+908)*a(n-2) +2*(-4*n^2+51*n-113)*a(n-3) -2*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022 EXAMPLE a(2)=14 because there are 10 paths from (0,0) to (6,0) (see A027307): u(d)u(d), u(d)Ud(d), uud(d), uUdd(d), Ud(d)u(d), Ud(d)Ud(d), Udud(d), UdUdd(d), Uudd(d) and UUddd(d), the fourteen returns to the x-axis being shown between parentheses. MAPLE a:=n->2+(1/n)*sum((3*n-j)*2^(n-j)*binomial(n, j)*binomial(2*n, n-j-1)/(n+j+2), j=0..n-2): seq(a(n), n=1..21); CROSSREFS Cf. A027307, A108435, A032349. Sequence in context: A051708 A355362 A074618 * A088754 A103945 A111713 Adjacent sequences: A108433 A108434 A108435 * A108437 A108438 A108439 KEYWORD nonn AUTHOR Emeric Deutsch, Jun 04 2005 STATUS approved

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Last modified September 24 13:15 EDT 2023. Contains 365579 sequences. (Running on oeis4.)