A002776 Terms in certain determinants.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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]

Crossrefs

Cf. A047922.

Programs

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

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 *)

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

Extensions

Edited by Dean Hickerson, Sep 20 2002