A002776 Terms in certain determinants. (Formerly M3972 N1642)

1, 5, 34, 258, 2136, 19320, 190800, 2051280, 23909760, 300827520, 4067884800, 58877280000, 908666035200, 14901260774400, 258832346572800, 4748165630208000, 91746433658880000, 1862735060938752000, 39649900359573504000, 883021783867711488000

Offset

0,2

Comments

a(n) equals (n+1)^2 times the permanent of the (n+1) X (n+1) matrix with 1/(n+1) in the top right corner, 1/(n+1) in the bottom left corner, and 1's everywhere else. - John M. Campbell, May 25 2011

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Muniru A Asiru, Table of n, a(n) for n = 0..100

J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.

J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]

Formula

a(n) = (n^3 + n^2 + 2*n + 1)*n!.

a(n) = (n+3)! - 5*(n+2)! + 6*(n+1)! - n!. - Umut Özer, Dec 26 2017

E.g.f.: (1 + x + 3*x^2 + x^3)/(1 - x)^4. - Stefano Spezia, Apr 17 2022

Maple

A002776 := [seq(factorial(n+3) - 5 * factorial(n+2) + 6 * factorial(n+1) - factorial(n), n=0..100)]; # Muniru A Asiru, Jan 15 2018

Mathematica

Table[(n^3+n^2+2n+1)n!, {n, 0, 30}] (* Harvey P. Dale, Oct 28 2011 *)

Prog

(GAP) A002776 := List([0..100], n -> Factorial(n+3) - 5 * Factorial(n+2) + 6 * Factorial(n+1) - Factorial(n)); # Muniru A Asiru, Jan 15 2018
(Magma) [(n^3+n^2+2*n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jan 19 2018

Crossrefs

Cf. A047922.

Sequence in context: A080503 A078284 A083987 * A333804 A081342 A248055

Adjacent sequences: A002773 A002774 A002775 * A002777 A002778 A002779

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Edited by Dean Hickerson, Sep 20 2002

Status

approved