A005323 Column of Motzkin triangle.

1, 4, 14, 44, 133, 392, 1140, 3288, 9438, 27016, 77220, 220584, 630084, 1800384, 5147328, 14727168, 42171849, 120870324, 346757334, 995742748, 2862099185, 8234447672, 23713180780, 68350541480, 197188167735, 569371325796

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

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

Links

• R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013.
• R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
• Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

Crossrefs

Cf. A026300.

A diagonal of triangle A020474.

Programs

Maple

A005323 := proc(n)
    if n <= 5 then
        op(n-2,[1,4,14]) ;
    else
        n*(2*n+1)*procname(n-1)+3*n*(n-1)*procname(n-2) ;
        %/(n+5)/(n-3) ;
    end if;
end proc:
seq(A005323(n),n=3..20) ; # R. J. Mathar, Aug 17 2022

Mathematica

a[3] = 1; a[4] = 4;
a[n_] := a[n] = (n(3(n-1) a[n-2] + (2n+1) a[n-1])) / ((n-3)(n+5));
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Jul 27 2018 *)

Maxima

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 */

Formula

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.

G.f.: z^3*M^4, where M is g.f. of Motzkin numbers (A001006).

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

(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

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

a(n) ~ 2 * 3^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 17 2019

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003