1,2

Number of ordered trees with n+1 edges, having root of even degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

The connection to Motzkin numbers comes from the Lagrange inversion formula. - Michael Somos, Oct 10 2003

Number of horizontal steps in all Motzkin paths of length n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003

Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD and UUDD, we have alltogether two UHD's (shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003

Number of ordered trees with n+1 edges, having exactly one leaf at even height. Number of Dyck path of semilength n+1, having exactly one peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd and uuudd(ud)d (here u=(1,1),d=(1,-1) and the unique peak at even height is shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004

a(n)=number of Dyck (n+1)-paths containing exactly one UDU. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 01 2004

a(n) = A111808(n,n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005

This is a kind of Motzkin transform of A059841 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A059841 generates 1,0,1,2,6,16,... that is 1,0 followed by this sequence here. [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).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

T. D. Noe, Table of n, a(n) for n=1..200

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Sum{T(k, k-1)}, k = 1, 2, ..., n, where T is the array defined in A025177.

G.f.: 2x/[1-2x-3x^2+(1-x)sqrt(1-2x-3x^2)] - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2002

E.g.f.: exp(x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002.

a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-3k,n-1],{k,0,Floor[(n-1)/3]}]. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

a(n)=n*sum{k=0..floor((n-1)/2), C(n-1,2k)*C(k)}, C(n)=A000108(n); a(n)=sum{k=0..floor((n-1)/2), (2k+1)*C(n,2k+1)*C(k)}; a(n)=sum{k=0..n-1, sum{j=0..floor(k/2), C(k,2j)*C(2j+1,j)}}; - Paul Barry (pbarry(AT)wit.ie), Feb 05 2007

a(n)=(A002426(n+1)-A002426(n))/2; - Paul Barry (pbarry(AT)wit.ie), May 22 2008

a(n)=n*A001006(n-1). [From Paul Barry (pbarry(AT)wit.ie), Oct 05 2009]

seq( sum('binomial(i, k)*binomial(i-k, k+1)', 'k'=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]

(PARI) a(n)=if(n<0, 0, polcoeff((1+x+x^2)^n, n-1))

(PARI) a(n)=if(n<0, 0, n++; n*polcoeff(serreverse(x/(1+x+x^2)+x*O(x^n)), n))

A diagonal of A027907. Cf. A002426.

a(n)=n*A001006(n-1).

Sequence in context: A055544 A126285 A026163 this_sequence A025266 A074403 A151391

Adjacent sequences: A005714 A005715 A005716 this_sequence A005718 A005719 A005720

nonn,easy

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

More terms from Erich Friedman (efriedma(AT)stetson.edu), Jun 01 2001