A052572 E.g.f. (1+2x-2x^2)/(1-x)^2.

1, 4, 10, 36, 168, 960, 6480, 50400, 443520, 4354560, 47174400, 558835200, 7185024000, 99632332800, 1482030950400, 23538138624000, 397533007872000, 7113748561920000, 134449847820288000, 2676192208994304000

Offset

0,2

Comments

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]

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

Links

Table of n, a(n) for n=0..19.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 515

Index entries for sequences related to factorial base representation

Formula

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

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

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

For n <= 1, a(n) = (n+1)^2, for n > 1, a(n) = (n+1)! + 2*n! - Antti Karttunen, Sep 24 2016

From Amiram Eldar, Nov 06 2020: (Start)

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

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

Maple

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

Mathematica

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 *)

Prog

(Scheme, two different implementations)
(define (A052572 n) (if (zero? n) 1 (* (+ 3 n) (A000142 n))))
(define (A052572 n) (if (<= n 1) (* (+ 1 n) (+ 1 n)) (+ (A000142 (+ 1 n)) (* 2 (A000142 n)))))
;; Antti Karttunen, Sep 24 2016

Crossrefs

Essentially twice A038720.

Cf. A000142.

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

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

Sequence in context: A345245 A197552 A378732 * A079725 A154152 A357051

Adjacent sequences: A052569 A052570 A052571 * A052573 A052574 A052575

Keyword

easy,nonn

Author

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

Status

approved