2,2

Number of returns (i.e. down steps hitting the x-axis) in all Motzkin paths of length n. E.g. a(4)=9 because in the nine Motzkin paths of length 4, HHHH, HHU(D), HU(D)H, HUH(D), U(D)HH, U(D)U(D), UH(D)H, UHH(D) and UUD(D), where H=(1,0), U=(1,1), D=(1,-1), we have alltogether nine down steps D hitting the x-axis (shown in parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003

Number of nonnegative H,U,D paths of length n that end at height 2. Bijection to the Deutsch manifestation above: turn the last U carrying the path up to height 2 into a D. This gives a Motzkin n-path with a marked return D. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

Number of Motzkin paths of length n+2, starting with a (1,1) step, ending with a (1,-1) step and touching the x-axis at least three times. Example: a(3)=3 because we have UDHUD, UDUHD and UHDUD, where H=(1,0), U=(1,1), D=(1,-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 27 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1, 2, ..., n, s(0) = 0, s(n) = 2.

G.f.=2(1-z-q)/(1-z+q)^2, where q=sqrt(1-2z-3z^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003

G.f.=z^2*M^3, where M=1+zM+z^2*M^2 is the g.f. of the Motzkin numbers (A001006). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 27 2006

a(n) = -sqrt(-3)*(-1)^n*(3*n*(13*n+27)*hypergeom([1/2, n],[1],4/3)-hypergeom([1/2, n+1],[1],4/3)*(41*n^2+115*n+60))/(2*(n+3)*(n+5)*(n+6)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 12 2009]

M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: ser:=series(z^2*M^3, z=0, 35): seq(coeff(ser, z, n), n=2..28); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 27 2006

Cf. A026300.

Cf. A001006.

A diagonal of triangle A020474.

Cf. A001006.

Sequence in context: A094292 A000242 A077846 this_sequence A103780 A098182 A007046

Adjacent sequences: A005319 A005320 A005321 this_sequence A005323 A005324 A005325

nonn,new

N. J. A. Sloane (njas(AT)research.att.com).