OFFSET
3,2
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.
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
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
STATUS
approved