A052778 - OEIS

%I #28 Aug 20 2022 06:39:25

%S 0,0,0,0,0,120,360,1680,10080,72576,604800,5702400,59875200,691891200,

%T 8717829120,118879488000,1743565824000,27360571392000,457312407552000,

%U 8109673360588800,152056375511040000,3005349539512320000,62444484876533760000,1360632459941314560000

%N E.g.f.: x^4*log(-1/(-1+x)).

%C Previous name was: A simple grammar.

%H Amiram Eldar, <a href="/A052778/b052778.txt">Table of n, a(n) for n = 0..450</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=735">Encyclopedia of Combinatorial Structures 735</a>.

%F E.g.f.: x^4*log(-1/(-1+x)).

%F Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, (-n^2+3*n+4)*a(n) + (n-3)*a(n+1)}.

%F a(n) = n! / (n-4) (n > 4). - _Olivier Gérard_, Jun 13 2001

%F Sum_{n>=5} 1/a(n) = 49/6 - 3*e. - _Amiram Eldar_, Oct 07 2020

%F Sum_{n>=5} (-1)^(n+1)/a(n) = 5/e - 11/6. - _Amiram Eldar_, Aug 20 2022

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

%t a[n_] := If[n < 5, 0, n!/(n - 4)]; Array[a, 20, 0] (* _Amiram Eldar_, Oct 07 2020 *)

%t With[{nn=30},CoefficientList[Series[x^4 Log[-1/(x-1)],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jun 28 2021 *)

%Y Cf. A052747, A052794, A052759.

%K easy,nonn

%O 0,6

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

%E New name using e.g.f. from _Vaclav Kotesovec_, Oct 07 2020