1,2

Number of Dyck paths of semilength n+2 having exactly one occurrence of UUU, where U=(1,1). E.g. a(2)=4 because we have UDUUUDDD, UUUDDDUD, UUUDDUDD and UUUDUDDD, where U=(1,1) and D=(1,-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

a(n)=number of Motzkin (2n+2)-paths whose longest basin has length n-1. A basin is a sequence of contiguous flatsteps preceded by a down step and followed by an up step. Example: a(2) counts FUDFUD, UDFUDF, UDFUFD, UFDFUD. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

a(n-2) = A111808(n,n-3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005

a(n)=total number of valleys (DUs) in all Motzkin (n+3)-paths. Example: a(2)=4 counts the valleys (indicated by *) in FUD*UD, UD*UDF, UD*UFD, UFD*UD; the remaining 17 Motzkin 5-paths contain no valleys. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

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, Trinomial Coefficient

G.f.: 2z/[1-4z+z^2+6z^3+(1-3z+2z^3)sqrt(1-2z-3z^2)] - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

E.g.f. : exp(x)BesselI(3, 2x) [0, 0, 0, 1, 4, 15..] - Paul Barry (pbarry(AT)wit.ie), Sep 21 2004

Cf. A014531, A014533.

First differences are in A025181.

Sequence in context: A026110 A056327 A026328 this_sequence A094705 A055218 A107307

Adjacent sequences: A014529 A014530 A014531 this_sequence A014533 A014534 A014535

nonn

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 05 2000