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A005323 Column of Motzkin triangle.

%I A005323 M3480 #63 Jan 17 2024 09:07:04
%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).
%H A005323 R. De Castro, A. L. Ramírez and J. L. Ramírez, <a href="http://arxiv.org/abs/1310.2449">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, arXiv preprint arXiv:1310.2449 [cs.DM], 2013.
%H A005323 R. Donaghey and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0097-3165(77)90020-6">Motzkin numbers</a>, J. Combin. Theory, Series A, 23 (1977), 291-301.
%H A005323 Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H A005323 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005323 Simon Plouffe, <a href="https://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
%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). - _Mark van Hoeij_, Nov 12 2009
%F A005323 (n + 5) (n - 3) a(n) = n  (2 n + 1) a(n - 1) + 3 n (n - 1) a(n - 2). - _Simon Plouffe_, Feb 09 2012, corrected for offset Aug 17 2022
%F A005323 a(n) = 4*sum(j=ceiling((n-3)/2)..n+1, C(j,2*j-n+3)*C(n+1,j))/(n+1). - _Vladimir Kruchinin_, Mar 17 2014
%F A005323 a(n) ~ 2 * 3^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 17 2019
%p A005323 A005323 := proc(n)
%p A005323     if n <= 5 then
%p A005323         op(n-2,[1,4,14]) ;
%p A005323     else
%p A005323         n*(2*n+1)*procname(n-1)+3*n*(n-1)*procname(n-2) ;
%p A005323         %/(n+5)/(n-3) ;
%p A005323     end if;
%p A005323 end proc:
%p A005323 seq(A005323(n),n=3..20) ; # _R. J. Mathar_, Aug 17 2022
%t A005323 a[3] = 1; a[4] = 4;
%t A005323 a[n_] := a[n] = (n(3(n-1) a[n-2] + (2n+1) a[n-1])) / ((n-3)(n+5));
%t A005323 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Jul 27 2018 *)
%o A005323 (Maxima)
%o A005323 a(n):=(4*sum(binomial(j,2*j-n+3)*binomial(n+1,j),j,ceiling((n-3)/2),n+1))/(n+1); /* _Vladimir Kruchinin_, Mar 18 2014 */
%Y A005323 Cf. A026300.
%Y A005323 A diagonal of triangle A020474.
%K A005323 nonn,easy
%O A005323 3,2
%A A005323 _N. J. A. Sloane_
%E A005323 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003