cp's OEIS Frontend

A001450 a(n) = binomial(5*n,2*n).

%I A001450 #79 Jun 27 2025 18:04:13
%S A001450 1,10,210,5005,125970,3268760,86493225,2319959400,62852101650,
%T A001450 1715884494940,47129212243960,1300853625660225,36052387482172425,
%U A001450 1002596421878664480,27963143931814663880,781879430625942976880,21910242651571684460050,615167304833936727234180
%N A001450 a(n) = binomial(5*n,2*n).
%H A001450 T. D. Noe, <a href="/A001450/b001450.txt">Table of n, a(n) for n=0..100</a>
%H A001450 Peter Bala, <a href="/A001450/a001450.txt">A note on A001450</a>
%H A001450 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 A001450 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 A001450 a(n) = (5*n)!/((3*n)!*(2*n)!).
%F A001450 a(n) = 2F1[-3n,-2n,1,1] (see Mathematica code below). - _John M. Campbell_, Jul 15 2011
%F A001450 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 A001450 From _Peter Bala_, Oct 05 2015: (Start)
%F A001450 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 A001450 a(n) = 5*A259550(n) for n >= 1.
%F A001450 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 A001450 a(n) = [x^(2*n)] 1/(1 - x)^(3*n+1). - _Ilya Gutkovskiy_, Oct 10 2017
%F A001450 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 A001450 a(n) = Sum_{k = 0..2*n} binomial(3*n+k-1, k). Cf. A066802. - _Peter Bala_, Jun 04 2024
%F A001450 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
%F A001450 From _Karol A. Penson_, May 07 2025: (Start)
%F A001450 G.f. denoted by h(x) satisfies the following algebraic equation of order 10:
%F A001450 8 - 3125*x + 20*(-13 + 3125*x)*h(x) - 45*(-74 + 9375*x)*h(x)^2 + 5*(-4023 + 175000*x)*h(x)^2 + 5*(-4023 + 175000*x)*h(x)^3 + 25*(1809 + 53125*x)*h(x)^4 + (34375*x - 738)*(3125*x - 108)*h(x)^5 + 15*(3125*x + 297)*(3125*x - 108)*h(x)^6 + 5*(3125*x - 108)^2*h(x)^7 + 135*(3125*x - 108)^2*h(x)^8 + (3125*x - 108)^3*h(x)^10=0.
%F A001450 a(n) = Integral_{x=0..3125/108} x^n*W(x)*dx, n>=0, where W(x) = W1(x)+W2(x)+W3(x)+W4(x) can be expressed with four generalized hypergeometric functions of type 4F3:
%F A001450 W1(x) = sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*hypergeom([1/5, 8/15, 7/10, 13/15], [2/5, 3/5, 4/5], (108*x)/3125)/(10*Pi*x^(4/5)),
%F A001450 W2(x) = sqrt(5)*sec((3*Pi)/10)*sin(Pi/10)*hypergeom([2/5, 11/15, 9/10, 16/15], [3/5, 4/5, 6/5], (108*x)/3125)/(50*Pi*x^(3/5)),
%F A001450 W3(x) = sqrt(5)*sec((3*Pi)/10)*sin(Pi/10)*hypergeom([3/5, 14/15, 11/10, 19/15], [4/5, 6/5, 7/5], (108*x)/3125)/(125*Pi*x^(2/5)), and
%F A001450 W4(x) = (7*sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*hypergeom([4/5, 17/15, 13/10, 22/15], [6/5, 7/5, 8/5], (108*x)/3125))/(1250*Pi*x^(1/5)).
%F A001450 Using the formula for a(n) only, W(x) can be shown to be a positive function. It is singular at x=0 and at x=3125/108. This integral representation is unique since W(x) is the solution of the Hausdorff moment problem. (End)
%F A001450 From _Peter Bala_, Jun 21 2025: (Start)
%F A001450 a(n) = [x^(3*n)] 1/(1 - x)^(2*n+1).
%F A001450 a(n) = Sum_{k = 0..3*n} binomial(2*n+k-1, k). (End)
%p A001450 f := n->(5*n)!/((3*n)!*(2*n)!);
%t A001450 Table[Hypergeometric2F1[-3n,-2n,1,1],{n,0,60}] (* _John M. Campbell_, Jul 15 2011 *)
%t A001450 Table[Binomial[5n,2n],{n,0,20}] (* _Harvey P. Dale_, Nov 09 2011 *)
%o A001450 (Magma) [Binomial(5*n, 2*n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%o A001450 (PARI) a(n) = binomial(5*n,2*n) \\ _Altug Alkan_, Oct 06 2015
%Y A001450 Cf. A001451, A001459, A001460, A000108, A060941, A066802, A259550.
%K A001450 nonn,easy
%O A001450 0,2
%A A001450 _N. J. A. Sloane_