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E.g.f. 1 - x - sqrt(1-4*x).
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%I #33 Jan 30 2020 21:29:14

%S 0,1,4,24,240,3360,60480,1330560,34594560,1037836800,35286451200,

%T 1340885145600,56317176115200,2590590101299200,129529505064960000,

%U 6994593273507840000,405686409863454720000,25152557411534192640000

%N E.g.f. 1 - x - sqrt(1-4*x).

%H G. C. Greubel, <a href="/A052718/b052718.txt">Table of n, a(n) for n = 0..365</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=674">Encyclopedia of Combinatorial Structures 674</a>

%F a(0)=0, a(1)=1, a(n) = 2^(2n-1) * Gamma(n-1/2) / sqrt(Pi) for n>=2. - _Jon E. Schoenfield_, Jan 11 2015

%F D-finite with recurrence: a(1)=1; a(2)=4; (2-4*n)*a(n) + a(n+1) = 0.

%F O.g.f.: 1-x-_2F_0(-1/2,1;;4*x). - _R. J. Mathar_, Feb 23 2010

%F G.f.: 1-x - G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k-2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 04 2013

%F a(n) = n!*A068875(n-1). - _R. J. Mathar_, Oct 18 2013

%p spec := [S,{B=Union(Z,C),C=Prod(B,B),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t s=2;lst={0, 1};Do[s+=n*s+s;AppendTo[lst, s], {n, 0, 5!, 4}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008 *)

%t With[{nn = 50}, CoefficientList[Series[1 - x - Sqrt[1 - 4*x], {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, Feb 16 2017 *)

%o (PARI) x='x+O('x^50); concat([0], Vec(serlaplace(1-x-sqrt(1-4*x)))) \\ _G. C. Greubel_, Feb 16 2017

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000