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 A110609 a(n) = n * C(2*n,n-1). 3

%I

%S 0,1,8,45,224,1050,4752,21021,91520,393822,1679600,7113106,29953728,

%T 125550100,524190240,2181340125,9051563520,37467344310,154754938800,

%U 637982011590,2625648168000,10789623755820,44277560801760,181478535620850,742984788858624

%N a(n) = n * C(2*n,n-1).

%C Second column of number triangle A110608.

%H G. C. Greubel, <a href="/A110609/b110609.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: -((2*x*(2*x+2*sqrt(1-4*x)-3) - sqrt(1-4*x) + 1)/(2*sqrt((1 - 4*x)^3)* x^2)). - _Marco A. Cisneros Guevara_, Jul 23 2011

%F (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

%F From _Ilya Gutkovskiy_, Jan 20 2017: (Start)

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

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

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

%p 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

%p 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

%t Table[CatalanNumber[n]*n^2, {n, 0, 22}] (* _Zerinvary Lajos_, Jul 08 2009 *)

%t 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 *)

%o (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

%o (PARI) for(n=0,25, print1(n*binomial(2*n,n-1), ", ")) \\ _G. C. Greubel_, Sep 01 2017

%Y Cf. A253487.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jul 30 2005

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Last modified July 20 15:59 EDT 2019. Contains 325185 sequences. (Running on oeis4.)