0,3

Number of ordered trees with n+1 edges, having root of degree 2 and nonroot nodes of outdegree at most 2.

Sequence without the initial 0 is the convolution of the sequence of Motzkin numbers (A001006) with itself.

Number of horizontal steps at level zero in all Motzkin paths of length n. Example: a(3)=5 because in the four Motzkin paths of length 3, (HHH), (H)UD, UD(H) and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have alltogether five horizontal steps H at level zero (shown in parentheses).

Number of peaks at level 1 in all Motzkin paths of length n+1. Example: a(3)=5 because in the nine Motzkin paths of length 4, HHHH, HH(UD), H(UD)H, HUHD, (UD)HH, (UD)(UD), UHDH, UHHD and UUDD (where H=(1,0), U=(1,1), D=(1,-1)), we have five peaks at level 1 (shown between parentheses).

a(n) = number of Motzkin paths of length n+1 that start with an up step. - David Callan (callan(AT)stat.wisc.edu), Jul 19 2004

Could be called a Motzkin transform of A130716 because the g.f. is obtained from the g.f. x*A130716(x)= x(1+x+x^2) (offset changed to 1) by the substitution x -> x*A001006(x) of the independent variable. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.

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

Sum(b = 1 to (n+1)/2) [ n choose (2b-1) ][ 2b choose b ]/(b+1).

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, Also T(n, n), where T is the array defined in A026105.

a(n)=sum{k=0..n-1, sum{i=0..k, C(k, 2i)*A000108(i+1) }}. - Paul Barry (pbarry(AT)wit.ie), Jul 18 2003

G.f.=4z/[1-z+sqrt(1-2z-3z^2)]^2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003.

a(n)=A005043(n+2)-A005043(n); - Paul Barry (pbarry(AT)wit.ie), Apr 17 2005

Cf. A001006, A026300, A026107.

A diagonal of triangle A020474.

Sequence in context: A051163 A051450 A038508 this_sequence A105695 A026938 A086622

Adjacent sequences: A002023 A002024 A002025 this_sequence A002027 A002028 A002029

nonn,easy,nice

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

Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003