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Number of sticky functions: endofunctions of [n] having a fixed point.
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%I #76 Aug 09 2024 01:56:03

%S 1,3,19,175,2101,31031,543607,11012415,253202761,6513215599,

%T 185311670611,5777672071535,195881901213181,7174630439858727,

%U 282325794823047151,11878335717996660991,532092356706983938321,25283323623228812584415,1270184310304975912766347

%N Number of sticky functions: endofunctions of [n] having a fixed point.

%C a(n) is also the number of functions f{1,2,...,n}->{1,2,...,n} with at least one element mapped to 1. - _Geoffrey Critzer_, Dec 10 2012

%C Equivalently, a(n) is the number of endofunctions with minimum 1. - _Olivier Gérard_, Aug 02 2016

%C Number of bargraphs of width n and height n. Equivalently: number of ordered n-tuples of positive integers such that the largest is n. Example: a(3) = 19 because we have 113, 123, 213, 223, 131, 132, 231, 232, 311, 312, 321, 322, 331, 332, 313, 323, 133, 233, and 333. - _Emeric Deutsch_, Jan 30 2017

%H Vincenzo Librandi, <a href="/A045531/b045531.txt">Table of n, a(n) for n = 1..100</a>

%H A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, <a href="http://dx.doi.org/10.1016/j.dam.2014.08.026">The height and width of bargraphs</a>, Discrete Applied Math. 180, (2015), 36-44.

%F a(n) = n^n - (n-1)^n.

%F E.g.f.: (T - x)/(T-T^2), where T=T(x) is Euler's tree function (see A000169).

%F With interpolated zeros, ceiling(n/2)^ceiling(n/2) - floor(n/2)^ceiling(n/2). - _Paul Barry_, Jul 13 2005

%F a(n) = A047969(n,n). - _Alford Arnold_, May 07 2005

%F a(n) = Sum_{i=1..n} binomial(n,i)*(i-1)^(i-1)*(n-i)^(n-i) = Sum_{i=1..n} binomial(n,i)*A000312(i-1)*A000312(n-i). - _Vladimir Shevelev_, Sep 30 2010

%F a(n) = Sum_{k=1..n} k!*binomial(n-1,k-1)*Stirling2(n,k). - _Vladimir Kruchinin_, Mar 01 2014

%F a(n) = A350454(n+1, 1) / (n+1). - _Mélika Tebni_, Dec 20 2022

%F Limit_{n->oo} a(n)/n^n = 1 - 1/e = A068996. - _Luc Rousseau_, Jan 20 2023

%t Table[Sum[Binomial[n, i] (n - 1)^(n - i), {i, 1, n}], {n, 1, 20}]

%o (Magma) [n^n-(n-1)^n: n in [1..20] ]; // _Vincenzo Librandi_, May 07 2011

%o (PARI) a(n)=n^n-(n-1)^n; \\ _Charles R Greathouse IV_, May 08 2011

%o (Maxima) a(n) = sum(k!*binomial(n-1,k-1)*stirling2(n,k),k,1,n); /* _Vladimir Kruchinin_, Mar 01 2014 */

%Y Column |a(n, 2)| of A039621. Row sums of triangle A055858.

%Y Cf. A000312, A066274, A066275, A047969.

%Y Column k=1 of A246049.

%Y Cf. A068996, A350454.

%K easy,nonn

%O 1,2

%A _Len Smiley_