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%I #79 Feb 23 2024 08:04:32
%S 1,6,36,220,1365,8568,54264,346104,2220075,14307150,92561040,
%T 600805296,3910797436,25518731280,166871334960,1093260079344,
%U 7174519270695,47153358767970,310325523515700,2044802197953900,13488561475572645,89067326568860640,588671286046028640
%N a(n) = binomial(3*n, n - 1).
%D Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%H Seiichi Manyama, <a href="/A004319/b004319.txt">Table of n, a(n) for n = 1..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%F G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - _Emeric Deutsch_, May 22 2003
%F a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - _Ralf Stephan_, Jun 03 2005
%F D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - _R. J. Mathar_, Feb 05 2013
%F a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - _Vladimir Kruchinin_, May 15 2013
%F G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - _R. J. Mathar_, Aug 10 2015
%F a(n) = n*A001764(n). - _R. J. Mathar_, Aug 10 2015
%F From _Peter Bala_, Nov 04 2015: (Start)
%F With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
%F G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - _Vladimir Kruchinin_, May 13 2016
%F a(n) = [x^(2*n+1)] 1/(1 - x)^n. - _Ilya Gutkovskiy_, Oct 10 2017
%F a(n) = binomial(n+1, 2) * A000139(n). - _F. Chapoton_, Feb 23 2024
%p A004319 := proc(n)
%p binomial(3*n,n-1);
%p end proc: # _R. J. Mathar_, Aug 10 2015
%t Table[Binomial[3n, n - 1], {n, 20}] (* _Harvey P. Dale_, Sep 21 2011 *)
%o (Maxima)
%o a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* _Vladimir Kruchinin_, May 15 2013 */
%o (PARI) vector(30, n, binomial(3*n, n-1)) \\ _Altug Alkan_, Nov 04 2015
%Y Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).
%Y Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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