%I A005323 M3480
%S A005323 1,4,14,44,133,392,1140,3288,9438,27016,77220,220584,630084,1800384,
%T A005323 5147328,14727168,42171849,120870324,346757334,995742748,2862099185,
%U A005323 8234447672,23713180780,68350541480,197188167735,569371325796
%N A005323 Column of Motzkin triangle.
%D A005323 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005323 R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series 
               A, 23 (1977), 291-301.
%F A005323 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative 
               integer and |s(i) - s(i-1)| <= 1 for i = 1, 2, ..., n, s(0) = 0, 
               s(n) = 3.
%F A005323 G.f.: z^3*M^4, where M is g.f. of Motzkin numbers (A001006).
%F A005323 a(n) = 4*(-3)^(1/2)*(-1)^n*n*((-3*n^3-9*n^2-6*n-9)*hypergeom([1/2, n],
               [1],4/3)+(2*n^3+n^2-17*n-13)*hypergeom([1/2, n+1],[1],4/3))/(3*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)) 
               (for n >= 3) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 12 
               2009]
%Y A005323 Cf. A026300.
%Y A005323 A diagonal of triangle A020474.
%Y A005323 Sequence in context: A006645 A094309 A000300 this_sequence A027831 A097894 
               A065835
%Y A005323 Adjacent sequences: A005320 A005321 A005322 this_sequence A005324 A005325 
               A005326
%K A005323 nonn,easy
%O A005323 3,2
%A A005323 N. J. A. Sloane (njas(AT)research.att.com).
%E A005323 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 
               29 2003