A101334 a(n) = n^n - (n+1)^(n-1).

0, 0, 1, 11, 131, 1829, 29849, 561399, 11994247, 287420489, 7642052309, 223394306387, 7123940054219, 246181194216957, 9165811757198641, 365836296342931439, 15584321022199735823, 705800730789742512401, 33866021217511735389485, 1716275655660313589123979

Offset: 0

Author

Jorge Coveiro, Dec 24 2004

Keywords

• nonn

Comments

b(n) = n^n mod (n+1)^(n-1) begins: 0, 0, 1, 11, 6, 533, 13042, 37111, 2428309, ... - Alex Ratushnyak, Aug 06 2012

a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} with at least one cycle of length >= 2. - Geoffrey Critzer, Jan 11 2013

Number of defective parking functions of length n and at least one defect. - Alois P. Heinz, Aug 18 2017

Examples

			a(3) = 3^3 - 4^2 = 27-16 = 11.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, and Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.
• Luca Ferrari and Francesco Verciani, On the enumeration of permutation-invariant and complete Naples parking functions, arXiv:2411.06876 [math.CO], 2024. See p. 11.

Crossrefs

Cf. A000272, A000312, A046065, A264902.

Programs

Mathematica

ReplacePart[Table[n^n-(n+1)^(n-1),{n,0,nn}],0,1]  (* Geoffrey Critzer, Jan 11 2013 *)

PARI

for(x=1,20,print( x^x-(x+1)^(x-1) ))

Python

print([n**n - (n+1)**(n-1) for n in range(33)]) # Alex Ratushnyak, Aug 06 2012

Formula

E.g.f.: 1/(1-T(x)) - exp(T(x)) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Jan 11 2013

a(n) = Sum_{k>0} A264902(n,k). - Alois P. Heinz, Nov 29 2015

a(n) = A000312(n) - A000272(n+1). - Alois P. Heinz, Aug 18 2017

Extensions

a(0) prepended by Alex Ratushnyak, Aug 06 2012