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%I M4176 #63 Apr 13 2022 13:25:17
%S 1,6,27,104,369,1242,4037,12804,39897,122694,373581,1128816,3390582,
%T 10136556,30192102,89662216,265640691,785509362,2319218869,6839057544,
%U 20147488020,59306494520,174466248840,512987904000,1507780192035,4430417492826,13015498076181
%N Column of Motzkin triangle.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005325/b005325.txt">Table of n, a(n) for n = 5..1000</a>
%H R. Donaghey and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0097-3165(77)90020-6">Motzkin numbers</a>, J. Combin. Theory, Series A, 23 (1977), 291-301.
%H Nickolas Hein, Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
%F G.f.: z^5*M^6, where M=1+z*M+z^2*M^2 is the g.f. for the Motzkin numbers (A001006). - _Emeric Deutsch_, Aug 13 2004
%F a(n) = (sqrt(-3)/81)*((-1)^n*n*(4*n^3-15*n^2-55*n+102)/(n+7)/(n+3)/(n+2)*hypergeom([1/2, n+7],[3],4/3)-(-1)^n*(4*n^4-17*n^3+23*n^2+ 242*n-288)/(n+7)/(n+3)/(n+2)*hypergeom([1/2, n+6],[3],4/3)). - _Mark van Hoeij_, Oct 29 2011.
%F a(n) (n + 11) (n - 1) = (n + 4) (3 n + 9) a(n - 2) + (n + 4) (2 n + 9) a(n - 1). - _Simon Plouffe_, Feb 09 2012
%F a(n) ~ 3^(n+5/2)/(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Oct 05 2012
%F a(n) = 6*sum(j=ceiling((n-5)/2)..(n+1), C(j,2*j-n+5)*C(n+1,j))/(n+1). - _Vladimir Kruchinin_, Mar 17 2014
%t RecurrenceTable[{3(-1+n)*n*a[-2+n]+n*(1+2n)*a[-1+n]-(-5+n)*(7+n)*a[n]==0, a[5]==1,a[6]==6}, a,{n,5,20}] (* _Vaclav Kotesovec_, Oct 05 2012 *)
%t a = DifferenceRoot[Function[{b, n}, {(-2n^2 - 25n - 78)b[n+1] - 3(n+5)(n+6) b[n] + (n+1)(n+13)b[n+2] == 0, b[1] == 1, b[2] == 6}]][# - 4]&;
%t Table[a[n], {n, 5, 31}] (* _Jean-François Alcover_, Jan 24 2019 *)
%o (Maxima)
%o a(n) := 6*sum(binomial(j,2*j-n+5)*binomial(n+1,j),j,ceiling((n-5)/2),(n+1))/(n+1);
%o /* _Vladimir Kruchinin_, Mar 18 2014 */
%Y Cf. A026300, A026126, A001006.
%Y A diagonal of triangle A020474.
%K nonn
%O 5,2
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, May 03 2013
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