A014532 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 3rd column from the center.

1, 4, 15, 50, 161, 504, 1554, 4740, 14355, 43252, 129844, 388752, 1161615, 3465840, 10329336, 30759120, 91538523, 272290140, 809676735, 2407049106, 7154586747, 21263575256, 63191778950, 187790510700, 558069593445, 1658498131836

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)

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

Cf. A014531, A014533.

First differences are in A025181.

Sequence in context: A026110 A056327 A026328 * A094705 A280786 A283276

Adjacent sequences: A014529 A014530 A014531 * A014533 A014534 A014535

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 05 2000

Status

approved