A005324 - OEIS

%I M3902 #61 Aug 17 2022 10:36:28

%S 1,5,20,70,230,726,2235,6765,20240,60060,177177,520455,1524120,

%T 4453320,12991230,37854954,110218905,320751445,933149470,2714401580,

%U 7895719634,22969224850,66829893650,194486929650,566141346225,1648500576021

%N Column of Motzkin triangle A026300.

%C 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 a(n) = T(n,n-4), where T is the array in A026300.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 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 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 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 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 G.f.: z^4*M^5, where M is g.f. of Motzkin numbers (A001006).

%F 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 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 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 a(n) = A026300(n,n-4). - _R. J. Mathar_, Aug 17 2022

%p A005324 := proc(n)

%p     if n <= 6 then

%p         op(n-3,[1,5,20]) ;

%p     else

%p         n*(2*n+1)*procname(n-1)+3*n*(n-1)*procname(n-2) ;

%p         %/(n+6)/(n-4) ;

%p     end if;

%p end proc:

%p seq(A005324(n),n=4..20) ; # _R. J. Mathar_, Aug 17 2022

%t T[n_, k_] := Sum[m = 2j+n-k; Binomial[n, m] (Binomial[m, j] - Binomial[m, j-1]), {j, 0, k/2}];

%t a[n_] := T[n, n-4];

%t Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, Jul 27 2018 *)

%o (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 Cf. A026300.

%Y A diagonal of triangle A020474.

%K nonn,easy

%O 4,2

%A _N. J. A. Sloane_