A052585 - OEIS

%I #23 Sep 08 2022 08:44:59

%S 1,1,6,30,264,2520,30960,428400,6894720,123742080,2478470400,

%T 54486432000,1308153369600,34005760588800,952248474777600,

%U 28566146568960000,914137612996608000,31080323154456576000,1118898035934142464000,42518003720397004800000

%N E.g.f. 1/(1-x-2*x^2).

%C Laguerre transform is A052563. - _Paul Barry_, Aug 08 2008

%H G. C. Greubel, <a href="/A052585/b052585.txt">Table of n, a(n) for n = 0..350</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=530">Encyclopedia of Combinatorial Structures 530</a>

%F E.g.f.: 1/(1 -x -2*x^2).

%F Recurrence: a(1)=1, a(0)=1, (-2*n^2-6*n-4)*a(n)+(-2-n)*a(n+1)+a(n+2)=0.

%F a(n) = Sum(1/9*(1+4*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+2*_Z^2))*n!.

%F a(n) = n!*A001045(n+1). - _Paul Barry_, Aug 08 2008

%F a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A080599 and A005442. - Peter Bala, Dec 07 2011

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

%t With[{m = 50}, CoefficientList[Series[-1/(-1 + x + 2*x^2), {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 17 2018 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1 -x -2*x^2))) \\ _G. C. Greubel_, May 17 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1 -x -2*x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 17 2018

%Y Cf. A080599, A005442.

%K easy,nonn

%O 0,3

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

%E a(18)-a(19) added by _G. C. Greubel_, May 17 2018