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A014532
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Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 3rd column from the center.
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (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
a(n) is the number of lattice paths from $(0,0)$ to \ $(n+1,n-1)$ avoiding $% \uparrow ^{\geq 3}$ [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Apr 20 2010]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
Shanzhen Gao, Pattern Avoidance in Paths and Walks, in preparation [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Apr 20 2010]
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..200
Eric Weisstein's World of Mathematics, Trinomial Coefficient
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FORMULA
| 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
a(n)=$\dsum\limits_{i=0}^{\lfloor (n-1)/2\rfloor }\binom{n+2}{n-1-i}\binom{n-1-i}{% i}$ [From Shanzhen Gao (shanzhengao(AT)yahoo.com), 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
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CROSSREFS
| Cf. A014531, A014533.
First differences are in A025181.
Sequence in context: A026110 A056327 A026328 * A094705 A196835 A055218
Adjacent sequences: A014529 A014530 A014531 * A014533 A014534 A014535
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 05 2000
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