This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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, 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. 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 + 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 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 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,changed AUTHOR EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 23 01:24 EDT 2018. Contains 316518 sequences. (Running on oeis4.)