cp's OEIS Frontend

A052717 Expansion of e.g.f. x*(1 - sqrt(1 - 4*x))/2.

%I A052717 #29 May 29 2022 03:16:44
%S A052717 0,0,2,6,48,600,10080,211680,5322240,155675520,5189184000,
%T A052717 194075481600,8045310873600,366061644748800,18134130709094400,
%U A052717 971471287987200000,55956746188062720000,3448334483839365120000
%N A052717 Expansion of e.g.f. x*(1 - sqrt(1 - 4*x))/2.
%H A052717 G. C. Greubel, <a href="/A052717/b052717.txt">Table of n, a(n) for n = 0..350</a>
%H A052717 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=673">Encyclopedia of Combinatorial Structures 673</a>
%F A052717 Recurrence: a(1)=0, a(3)=6, a(2)=2, n*a(n+1) = (4*n^2 - 2*n - 6)*a(n).
%F A052717 a(n) = n!*A000108(n-2) = A052711(n), n > 2. - _R. J. Mathar_, Oct 26 2013
%F A052717 G.f.: x*(d/dx)(x^2 * Hypergeometric2F0([1, 1/2], [], 4*x)). - _G. C. Greubel_, May 28 2022
%p A052717 spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052717 with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Permutation(n*2-2),size=n-1)),j=0..n) od: seq(a[n], n=0..22); # _Zerinvary Lajos_, May 03 2007
%t A052717 With[{nn=20},CoefficientList[Series[x (1-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 20 2015 *)
%t A052717 Table[Boole[n==1] + n!*CatalanNumber[n-2], {n, 0, 30}] (* _G. C. Greubel_, May 28 2022 *)
%o A052717 (MuPAD) combinat::catalan(n)*(n+2)! $ n = 0..15; // _Zerinvary Lajos_, Feb 15 2007
%o A052717 (Magma) [n le 1 select 0 else Factorial(n)*Catalan(n-2): n in [0..30]]; // _G. C. Greubel_, May 28 2022
%o A052717 (SageMath) [bool(n==1)/2 + factorial(n)*catalan_number(n-2) for n in (0..30)] # _G. C. Greubel_, May 28 2022
%Y A052717 Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052718, A052719, A052720, A052721, A052722, A052723.
%Y A052717 Cf. A000108, A052711.
%K A052717 easy,nonn
%O A052717 0,3
%A A052717 encyclopedia(AT)pommard.inria.fr, Jan 25 2000