A110609 a(n) = n * binomial(2*n, n-1).

0, 1, 8, 45, 224, 1050, 4752, 21021, 91520, 393822, 1679600, 7113106, 29953728, 125550100, 524190240, 2181340125, 9051563520, 37467344310, 154754938800, 637982011590, 2625648168000, 10789623755820, 44277560801760, 181478535620850, 742984788858624, 3038716500907500

Offset: 0

Author

Paul Barry, Jul 30 2005

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Crossrefs

Cf. A000108, A002390, A002736, A253487.

Column k=1 of A110608.

Programs

Magma

[0] cat [((4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2))/2: n in [0..25]]; // Vincenzo Librandi, Jan 09 2015

Maple

with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1),size=n)),j=1..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 09 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1), j=1..n),k=1..n): seq(a(n), n=0..22); # Zerinvary Lajos, May 09 2007
series(simplify(x*diff(x*diff((1-sqrt(1-4*x))/(2*x), x), x)), x, 20):
seq(coeff(%, x, k), k=0..18); # Karol A. Penson, Apr 25 2025

Mathematica

Table[CatalanNumber[n]*n^2, {n, 0, 22}] (* Zerinvary Lajos, Jul 08 2009 *)
CoefficientList[Series[x (1 / x^2 - (1 - 6 x + 4 x^2) / ((1 - 4 x)^(3/2) x^2)) / 2, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 09 2015 *)

PARI

for(n=0,25, print1(n*binomial(2*n,n-1), ", ")) \\ G. C. Greubel, Sep 01 2017

Formula

a(n) = n^2*binomial(2*n, n)/(n+1) = n^2*A000108(n) = A002736(n)/(n+1).

G.f.: -(2*x*(2*x+2*sqrt(1-4*x)-3) - sqrt(1-4*x) + 1)/(2*sqrt((1 - 4*x)^3)*x). - Marco A. Cisneros Guevara, Jul 23 2011; amended by Georg Fischer, Apr 09 2020

(n+1)*(10*n-7)*a(n)+2*n*(5*n-88)*a(n-1) -4*(25*n-22)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 07 2012

From Ilya Gutkovskiy, Jan 20 2017: (Start)

E.g.f.: x*(BesselI(0,2*x) + 2*BesselI(1,2*x) + BesselI(2,2*x))*exp(2*x).

a(n) ~ 4^n*sqrt(n)/sqrt(Pi).

Sum_{n>=1} 1/a(n) = Pi*(2*sqrt(3) + Pi)/18 = 1.152911143694148... (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (2/sqrt(5))*log(phi) + 2*log(phi)^2, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021

G.f.: (x*(d/dx))^2 [g.f. of A000108]. - Karol A. Penson, Apr 25 2025