A052572 - OEIS

%I #30 Nov 06 2020 03:52:49

%S 1,4,10,36,168,960,6480,50400,443520,4354560,47174400,558835200,

%T 7185024000,99632332800,1482030950400,23538138624000,397533007872000,

%U 7113748561920000,134449847820288000,2676192208994304000

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

%C a(n) equals the permanent of the (n+1) X (n+1) matrix whose entry directly below the entry in the top right corner is 3, and all of whose other entries are 1. [From _John M. Campbell_, May 25 2011]

%C In factorial base representation (A007623) the terms are written as: 1, 20, 120, 1200, 12000, 120000, ... From a(2) = 10 = "120" onward each term begins always with "120", followed by n-2 additional zeros. - _Antti Karttunen_, Sep 24 2016

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=515">Encyclopedia of Combinatorial Structures 515</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

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

%F Recurrence: {a(0)=1, a(1)=4, a(2)=10, (-n^2-5*n-4)*a(n)+(n+3)*a(n+1)=0}

%F a(n) = (n+3)*n! for n>0.

%F For n <= 1, a(n) = (n+1)^2, for n > 1, a(n) = (n+1)! + 2*n! - _Antti Karttunen_, Sep 24 2016

%F From _Amiram Eldar_, Nov 06 2020: (Start)

%F Sum_{n>=0} 1/a(n) = e - 4/3.

%F Sum_{n>=0} (-1)^n/a(n) = 8/3 - 5/e. (End)

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

%t With[{nn=20},CoefficientList[Series[(1+2x-2x^2)/(1-x)^2,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jul 03 2020 *)

%o (Scheme, two different implementations)

%o (define (A052572 n) (if (zero? n) 1 (* (+ 3 n) (A000142 n))))

%o (define (A052572 n) (if (<= n 1) (* (+ 1 n) (+ 1 n)) (+ (A000142 (+ 1 n)) (* 2 (A000142 n)))))

%o ;; _Antti Karttunen_, Sep 24 2016

%Y Essentially twice A038720.

%Y Cf. A000142.

%Y Row 7 of A276955, from a(2)=10 onward.

%Y Cf. sequences with formula (n + k)*n! listed in A282466.

%K easy,nonn

%O 0,2

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