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A005323
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Column of Motzkin triangle.
(Formerly M3480)
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6
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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; internal format)
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OFFSET
| 3,2
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REFERENCES
| R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, < a href="http://arxiv.org/ftp/arxiv/papers/0912/0912.0072.pdf"> Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), 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]
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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
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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