A052715 - OEIS

%I #21 May 28 2022 04:01:34

%S 0,0,0,0,24,480,10080,241920,6652800,207567360,7264857600,

%T 282291609600,12067966310400,563171761152000,28496491114291200,

%U 1554354060779520000,90929712555601920000,5679609738088366080000

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

%H G. C. Greubel, <a href="/A052715/b052715.txt">Table of n, a(n) for n = 0..350</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=671">Encyclopedia of Combinatorial Structures 671</a>

%F D-finite with recurrence: a(1)=0; a(2)=0; a(3)=0; a(4)=24; 4*(-n-3+2*n^2)*a(n) +2*(-1-3*n)*a(n+1) +a(n+2) =0.

%F a(n) = n!*A002057(n-4). - _R. J. Mathar_, Oct 18 2013

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

%t With[{nn=20},CoefficientList[Series[(1-2x-Sqrt[1-4x])/2-x(1-2x- Sqrt[ 1-4x])- x^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 18 2016 *)

%o (Magma) [n le 3 select 0 else 2*(n-3)*Factorial(n-1)*Catalan(n-2): n in [0..30]]; // _G. C. Greubel_, May 27 2022

%o (SageMath) [0]+[4*factorial(n-1)*binomial(2*n-5, n-4) for n in (1..30)] # _G. C. Greubel_, May 27 2022

%Y Cf. A000108, A002057.

%K easy,nonn

%O 0,5

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