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%I #46 May 13 2022 19:18:14
%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>
%H Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%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). - _Marco A. Cisneros Guevara_, Jul 23 2011; amended by _Georg Fischer_, Apr 09 2020
%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)
%F 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
%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. A002390, A253487.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jul 30 2005
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