A213326 - OEIS

A213326

a(n) = (n+2)^n - (n+1)^n.

0, 1, 7, 61, 671, 9031, 144495, 2685817, 56953279, 1357947691, 35979939623, 1049152023349, 33395827252815, 1152480295105231, 42864668012537311, 1709501546902968817, 72778339220927383295, 3294475298046105653971, 158016649702088758467159

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OFFSET

0,3

COMMENTS

a(n) is the number of acyclic functions from subsets of size n-1 or less of {1,...,n+1} to {1,2,...,n+1}. - Dennis P. Walsh, Nov 06 2015

a(n) is the number of parking functions whose largest element is not n+1 and length is n+1. For example, a(2) = 7 because there are seven such parking functions, namely (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,2), (2,1,2), (2,2,1). - Ran Pan, Nov 15 2015

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..200

FORMULA

a(n) = Sum_{i=0..n-1} binomial(n+1,i)*(i+1)^(i-1)*(n-i)^(n-i).

a(n) = A000272(n+2) - A000169(n+1).

E.g.f.: LambertW(-x)*(LambertW(-x)+x)/((1+LambertW(-x))*x^2). - Alois P. Heinz, Aug 12 2017

MAPLE

A213326:=n->(n+2)^n - (n+1)^n: seq(A213326(n), n=0..20); # Wesley Ivan Hurt, Nov 12 2015

MATHEMATICA

Table[(n + 2)^n - (n + 1)^n, {n, 0, 20}] (* T. D. Noe, Mar 07 2013 *)

PROG

(Maxima) a(n):=sum(binomial(n+1, i)*((i+1)^(i-1)*(n-i)^(n-i)), i, 0, n-1);