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

1, 10, 210, 5005, 125970, 3268760, 86493225, 2319959400, 62852101650, 1715884494940, 47129212243960, 1300853625660225, 36052387482172425, 1002596421878664480, 27963143931814663880, 781879430625942976880, 21910242651571684460050, 615167304833936727234180

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n=0..100

Peter Bala, A note on A001450

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.

Formula

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

a(n) = 2F1[-3n,-2n,1,1] (see Mathematica code below). - John M. Campbell, Jul 15 2011

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

From Peter Bala, Oct 05 2015: (Start)

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.

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

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)

a(n) = [x^(2*n)] 1/(1 - x)^(3*n+1). - Ilya Gutkovskiy, Oct 10 2017

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

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

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

Maple

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

Mathematica

Table[Hypergeometric2F1[-3n, -2n, 1, 1], {n, 0, 60}] (* John M. Campbell, Jul 15 2011 *)
Table[Binomial[5n, 2n], {n, 0, 20}] (* Harvey P. Dale, Nov 09 2011 *)

Prog

(Magma) [Binomial(5*n, 2*n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
(PARI) a(n) = binomial(5*n, 2*n) \\ Altug Alkan, Oct 06 2015

Crossrefs

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

Sequence in context: A334537 A052245 A052246 * A076803 A120596 A238467

Adjacent sequences: A001447 A001448 A001449 * A001451 A001452 A001453

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved