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A005324 Column of Motzkin triangle A026300.

%I A005324 M3902 #61 Aug 17 2022 10:36:28
%S A005324 1,5,20,70,230,726,2235,6765,20240,60060,177177,520455,1524120,
%T A005324 4453320,12991230,37854954,110218905,320751445,933149470,2714401580,
%U A005324 7895719634,22969224850,66829893650,194486929650,566141346225,1648500576021
%N A005324 Column of Motzkin triangle A026300.
%C A005324 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) = 4. - _Clark Kimberling_
%C A005324 a(n) = T(n,n-4), where T is the array in A026300.
%D A005324 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005324 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 A005324 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 A005324 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 A005324 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 A005324 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 A005324 G.f.: z^4*M^5, where M is g.f. of Motzkin numbers (A001006).
%F A005324 a(n) = (-5*I*(-1)^n*(n^4-6*n^3-43*n^2-24*n+36)*3^(1/2)*hypergeom([1/2, n+2],[1],4/3)+15*I*(-1)^n*(n^4+6*n^3+17*n^2+24*n-12)*3^(1/2)*hypergeom([1/2, n+1],[1],4/3))/(6*(n+3)*(n+2)*(n+4)*(n+5)*(n+6)). - _Mark van Hoeij_, Oct 29 2011
%F A005324 a(n) (n + 6) (n - 4) = 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 A005324 a(n) =  5*sum(j=ceiling((n-4)/2)..(n+1), binomial(j,2*j-n+4)*binomial(n+1,j))/(n+1). - _Vladimir Kruchinin_, Mar 17 2014
%F A005324 a(n) = A026300(n,n-4). - _R. J. Mathar_, Aug 17 2022
%p A005324 A005324 := proc(n)
%p A005324     if n <= 6 then
%p A005324         op(n-3,[1,5,20]) ;
%p A005324     else
%p A005324         n*(2*n+1)*procname(n-1)+3*n*(n-1)*procname(n-2) ;
%p A005324         %/(n+6)/(n-4) ;
%p A005324     end if;
%p A005324 end proc:
%p A005324 seq(A005324(n),n=4..20) ; # _R. J. Mathar_, Aug 17 2022
%t A005324 T[n_, k_] := Sum[m = 2j+n-k; Binomial[n, m] (Binomial[m, j] - Binomial[m, j-1]), {j, 0, k/2}];
%t A005324 a[n_] := T[n, n-4];
%t A005324 Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, Jul 27 2018 *)
%o A005324 (Maxima) a(n) := 5*sum(binomial(j,2*j-n+4)*binomial(n+1,j),j,ceiling((n-4)/2),(n+1))/(n+1); /* _Vladimir Kruchinin_, Mar 18 2014 */
%Y A005324 Cf. A026300.
%Y A005324 A diagonal of triangle A020474.
%K A005324 nonn,easy
%O A005324 4,2
%A A005324 _N. J. A. Sloane_