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A005323 Column of Motzkin triangle.
(Formerly M3480)
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

REFERENCES

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

LINKS

Table of n, a(n) for n=3..28.

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, 2013

R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. 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 + 7) (n - 1) a(n) = (n + 2) (2 n + 5) a(n - 1) + (n + 2) (3 n + 3) a(n - 2). - Simon Plouffe, Feb 09 2012

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

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

Cf. A026300.

A diagonal of triangle A020474.

Sequence in context: A006645 A094309 A000300 * A027831 A097894 A065835

Adjacent sequences:  A005320 A005321 A005322 * A005324 A005325 A005326

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified February 19 14:15 EST 2018. Contains 299333 sequences. (Running on oeis4.)