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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.

%I A014532 #78 Jul 02 2025 16:01:55
%S A014532 1,4,15,50,161,504,1554,4740,14355,43252,129844,388752,1161615,
%T A014532 3465840,10329336,30759120,91538523,272290140,809676735,2407049106,
%U A014532 7154586747,21263575256,63191778950,187790510700,558069593445,1658498131836
%N 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.
%C A014532 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
%C A014532 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
%C A014532 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
%C A014532 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
%C A014532 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
%D A014532 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%H A014532 G. C. Greubel, <a href="/A014532/b014532.txt">Table of n, a(n) for n = 1..1000</a> (terms 0..200 from T. D. Noe)
%H A014532 Mark Shattuck, <a href="https://doi.org/10.54550/ECA2024V4S4R32">Subword Patterns in Smooth Words</a>, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 10.
%H A014532 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%F A014532 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
%F A014532 E.g.f.: exp(x)*BesselI(3, 2x) [0, 0, 0, 1, 4, 15..]. - _Paul Barry_, Sep 21 2004
%F A014532 a(n-2) = A111808(n,n-3) for n>2. - _Reinhard Zumkeller_, Aug 17 2005
%F A014532 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
%F A014532 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
%F A014532 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
%F A014532 a(n) ~ 3^(n+5/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 10 2013
%F A014532 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
%F A014532 From _Peter Luschny_, May 09 2016: (Start)
%F A014532 a(n) = C(4+2*n, n-1)*hypergeom([-n+1, -n-5], [-3/2-n], 1/4).
%F A014532 a(n) = GegenbauerC(n-1, -n-2, -1/2).  (End)
%p A014532 a := n -> simplify(GegenbauerC(n-1, -n-2, -1/2)):
%p A014532 seq(a(n), n=1..26); # _Peter Luschny_, May 09 2016
%t A014532 Table[GegenbauerC[n - 1, -n - 2, -1/2], {n,1,50}] (* _G. C. Greubel_, Feb 28 2017 *)
%o A014532 (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
%Y A014532 Cf. A014531, A014533.
%Y A014532 First differences are in A025181.
%K A014532 nonn
%O A014532 1,2
%A A014532 _N. J. A. Sloane_
%E A014532 More terms from _James Sellers_, Feb 05 2000