A045531 Number of sticky functions: endofunctions of [n] having a fixed point.

1, 3, 19, 175, 2101, 31031, 543607, 11012415, 253202761, 6513215599, 185311670611, 5777672071535, 195881901213181, 7174630439858727, 282325794823047151, 11878335717996660991, 532092356706983938321, 25283323623228812584415, 1270184310304975912766347

Offset: 1

Author

Len Smiley

Keywords

• easy
• nonn

Comments

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

Equivalently, a(n) is the number of endofunctions with minimum 1. - Olivier Gérard, Aug 02 2016

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Crossrefs

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

Cf. A000312, A066274, A066275, A047969.

Column k=1 of A246049.

Cf. A068996, A350454.

Programs

Magma

[n^n-(n-1)^n: n in [1..20] ]; // Vincenzo Librandi, May 07 2011

Mathematica

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

Maxima

a(n) = sum(k!*binomial(n-1,k-1)*stirling2(n,k),k,1,n); /* Vladimir Kruchinin, Mar 01 2014 */

PARI

a(n)=n^n-(n-1)^n; \\ Charles R Greathouse IV, May 08 2011

Formula

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

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

With interpolated zeros, ceiling(n/2)^ceiling(n/2) - floor(n/2)^ceiling(n/2). - Paul Barry, Jul 13 2005

a(n) = A047969(n,n). - Alford Arnold, May 07 2005

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

a(n) = Sum_{k=1..n} k!*binomial(n-1,k-1)*Stirling2(n,k). - Vladimir Kruchinin, Mar 01 2014

a(n) = A350454(n+1, 1) / (n+1). - Mélika Tebni, Dec 20 2022

Limit_{n->oo} a(n)/n^n = 1 - 1/e = A068996. - Luc Rousseau, Jan 20 2023