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%I #32 Sep 08 2022 08:46:02
%S 0,1,7,61,671,9031,144495,2685817,56953279,1357947691,35979939623,
%T 1049152023349,33395827252815,1152480295105231,42864668012537311,
%U 1709501546902968817,72778339220927383295,3294475298046105653971,158016649702088758467159
%N a(n) = (n+2)^n - (n+1)^n.
%C 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
%C 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
%H G. C. Greubel, <a href="/A213326/b213326.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{i=0..n-2} binomial(n,i)*((i+1)^(i-1)*(n-i-1)^(n-i-1))).
%F a(n) = A000272(n+2) - A000169(n+1).
%F E.g.f.: LambertW(-x)*(LambertW(-x)+x)/((1+LambertW(-x))*x^2). - _Alois P. Heinz_, Aug 12 2017
%p A213326:=n->(n+2)^n - (n+1)^n: seq(A213326(n), n=0..20); # _Wesley Ivan Hurt_, Nov 12 2015
%t Table[(n + 2)^n - (n + 1)^n, {n, 0, 20}] (* _T. D. Noe_, Mar 07 2013 *)
%o (Maxima) a(n):=sum(binomial(n,i)*((i+1)^(i-1)*(n-i-1)^(n-i-1)),i,0,n-2);
%o (PARI) vector(40, n, n--; (n+2)^n-(n+1)^n) \\ _Altug Alkan_, Nov 11 2015
%o (Magma) [(n+2)^n - (n+1)^n : n in [0..20]]; // _Wesley Ivan Hurt_, Nov 12 2015
%Y Cf. A000169, A000272.
%K nonn,easy
%O 0,3
%A _Vladimir Kruchinin_, Mar 03 2013
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