OFFSET
1,2
COMMENTS
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, Dec 05 2003
a(n) is the 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, Jul 15 2004
a(n) is the 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, Jul 03 2006
a(n) is the number of lattice paths from (0,0) to (n+1,n-1) taking north and east steps avoiding north^{>=3}. - Shanzhen Gao, Apr 20 2010
a(n) is the number of paths in the half-plane x>=0, from (0,0) to (n+2,3), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=2, we have the 4 paths: HUUU, UHUU, UUHU, UUUH. - José Luis Ramírez Ramírez, Apr 19 2015
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 0..200 from T. D. Noe)
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 10.
Eric Weisstein's World of Mathematics, Trinomial Coefficient
FORMULA
G.f.: 2*z/(1-4*z+z^2+6*z^3+(1-3*z+2*z^3)*sqrt(1-2*z-3*z^2)). - Emeric Deutsch, Dec 05 2003
E.g.f.: exp(x)*BesselI(3, 2x) [0, 0, 0, 1, 4, 15..]. - Paul Barry, Sep 21 2004
a(n-2) = A111808(n,n-3) for n>2. - Reinhard Zumkeller, Aug 17 2005
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n+2,n-1-i) * binomial(n-1-i,i). - Shanzhen Gao, Apr 20 2010
a(n) = -(1/(162*(n+5)*(n+3)))*(9*n+18)*(-1)^n*(-3)^(1/2) * ((n+7)*hypergeom([1/2, n+5],[1],4/3) + hypergeom([1/2, n+4],[1],4/3) * (5*n+19)). - Mark van Hoeij, Oct 30 2011
D-finite with recurrence -(n+5)*(n-1)*a(n) +(n+2)*(2*n+3)*a(n-1) +3*(n+2)*(n+1)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
a(n) ~ 3^(n+5/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 10 2013
G.f.: z*M(z)^3/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths (A001006). - José Luis Ramírez Ramírez, Apr 19 2015
From Peter Luschny, May 09 2016: (Start)
a(n) = C(4+2*n, n-1)*hypergeom([-n+1, -n-5], [-3/2-n], 1/4).
a(n) = GegenbauerC(n-1, -n-2, -1/2). (End)
MAPLE
a := n -> simplify(GegenbauerC(n-1, -n-2, -1/2)):
seq(a(n), n=1..26); # Peter Luschny, May 09 2016
MATHEMATICA
Table[GegenbauerC[n - 1, -n - 2, -1/2], {n, 1, 50}] (* G. C. Greubel, Feb 28 2017 *)
PROG
(PARI) z='z+O('z^50); Vec(2*z/(1-4*z+z^2+6*z^3+(1-3*z+2*z^3)*sqrt(1-2*z-3*z^2))) \\ G. C. Greubel, Feb 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Feb 05 2000
STATUS
approved