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A005323
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Column of Motzkin triangle.
(Formerly M3480)
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7
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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
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OFFSET
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3,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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));
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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STATUS
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approved
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