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a(n) = binomial(5*n,2*n).
9

%I #62 Jun 12 2024 17:34:46

%S 1,10,210,5005,125970,3268760,86493225,2319959400,62852101650,

%T 1715884494940,47129212243960,1300853625660225,36052387482172425,

%U 1002596421878664480,27963143931814663880,781879430625942976880,21910242651571684460050,615167304833936727234180

%N a(n) = binomial(5*n,2*n).

%H T. D. Noe, <a href="/A001450/b001450.txt">Table of n, a(n) for n=0..100</a>

%H Peter Bala, <a href="/A001450/a001450.txt">A note on A001450</a>

%H M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

%H M. Dziemianczuk, <a href="https://doi.org/10.1016/j.disc.2015.11.001">On Directed Lattice Paths With Additional Vertical Steps</a>, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.

%F a(n) = (5*n)!/((3*n)!*(2*n)!).

%F a(n) = 2F1[-3n,-2n,1,1] (see Mathematica code below). - _John M. Campbell_, Jul 15 2011

%F G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (3125/108)*x). - _Robert Israel_, Aug 07 2014

%F From _Peter Bala_, Oct 05 2015: (Start)

%F a(n) = [x^n] ( (1 + x)*C(x) )^(5*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.

%F a(n) = 5*A259550(n) for n >= 1.

%F exp( (1/5) * Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. (End)

%F a(n) = [x^(2*n)] 1/(1 - x)^(3*n+1). - _Ilya Gutkovskiy_, Oct 10 2017

%F D-finite with recurrence 6*n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021

%F a(n) = Sum_{k = 0..2*n} binomial(3*n+k-1, k). Cf. A066802. - _Peter Bala_, Jun 04 2024

%F Right-hand side of the identity Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)* binomial(4*n-k, 2*n-k) = binomial(5*n, 2*n). Compare with the identity Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(4*n-k, 2*n-k) = binomial(3*n, n). - _Peter Bala_, Jun 05 2024

%p f := n->(5*n)!/((3*n)!*(2*n)!);

%t Table[Hypergeometric2F1[-3n,-2n,1,1],{n,0,60}] (* _John M. Campbell_, Jul 15 2011 *)

%t Table[Binomial[5n,2n],{n,0,20}] (* _Harvey P. Dale_, Nov 09 2011 *)

%o (Magma) [Binomial(5*n, 2*n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014

%o (PARI) a(n) = binomial(5*n,2*n) \\ _Altug Alkan_, Oct 06 2015

%Y Cf. A001451, A001459, A001460, A000108, A060941, A066802, A259550.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_