%I #26 May 28 2022 04:02:36

%S 0,2,4,36,528,10800,283680,9102240,345058560,15090727680,747888422400,

%T 41422381862400,2535569103513600,169983582318950400,

%U 12386182292118835200,974723523832041984000,82385641026424479744000

%N Expansion of e.g.f. (x + 1 - sqrt(1-6*x+x^2))/2.

%C With a(n)=1, also number of labeled mobiles with n nodes and 2-colored internal (non-leaf) nodes - _Christian G. Bower_, Jun 07 2005

%H G. C. Greubel, <a href="/A052716/b052716.txt">Table of n, a(n) for n = 0..345</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=672">Encyclopedia of Combinatorial Structures 672</a>

%H <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>

%F D-finite with recurrence: a(2)=4, a(1)=2, (n^2-1)*a(n) = (3+6*n)*a(n+1) - a(n+2).

%F a(n) = n!*A006318(n-1), n>=2. - _R. J. Mathar_, Oct 26 2013

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

%t With[{nn=20},CoefficientList[Series[(x+1-Sqrt[1-6x+x^2])/2,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Apr 19 2020 *)

%o (Magma) [n le 1 select 1-(-1)^n else Factorial(n)*(&+[Catalan(k)*Binomial(n+k-1, n-k-1): k in [0..n-1]]): n in [0..30]]; // _G. C. Greubel_, May 28 2022

%o (SageMath) [bool(n==1)+factorial(n)*sum(binomial(n+k-1, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # _G. C. Greubel_, May 28 2022

%Y Cf. A006318, A108531.

%K easy,nonn

%O 0,2

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