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)lhsystems.com), Aug 17 2005

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

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 A098617

Adjacent sequences: A005714 A005715 A005716 this_sequence A005718 A005719 A005720

nonn,easy

njas

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