A052671 - OEIS

%I #19 Jun 14 2022 01:44:40

%S 0,0,0,6,24,240,2880,40320,645120,11612160,232243200,5109350400,

%T 122624409600,3188234649600,89270570188800,2678117105664000,

%U 85699747381248000,2913791410962432000,104896490794647552000

%N Expansion of e.g.f. x^3*(1-x)/(1-2*x).

%H G. C. Greubel, <a href="/A052671/b052671.txt">Table of n, a(n) for n = 0..350</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=619">Encyclopedia of Combinatorial Structures 619</a>

%F E.g.f.: x^3*(-1+x)/(-1+2*x)

%F Recurrence: a(0)=0, a(1)=0, a(2)=0, a(3)=6, a(4)=24, a(n) = 2*n*a(n-1).

%F a(n) = 2^(n-4)*n!, n>3.

%F G.f.: 6*x^3 + 24*x^4*Hypergeometric2F0([5,1], [], 2*x). - _G. C. Greubel_, Jun 13 2022

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

%t With[{nn=20},CoefficientList[Series[x^3(-1+x)/(-1+2x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 13 2014 *)

%o (Magma) [0,0,0,6] cat [2^(n-4)*Factorial(n): n in [4..30]]; // _G. C. Greubel_, Jun 13 2022

%o (SageMath) [(2^(n-4) - 2^(n-4)*bool(n<3) + (1/2)*bool(n==3))*factorial(n) for n in (0..30)] # _G. C. Greubel_, Jun 13 2022

%Y Cf. A000079, A000142.

%K easy,nonn

%O 0,4

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