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 A045531 Number of sticky functions: endofunctions of [n] having a fixed point. 16
 1, 3, 19, 175, 2101, 31031, 543607, 11012415, 253202761, 6513215599, 185311670611, 5777672071535, 195881901213181, 7174630439858727, 282325794823047151, 11878335717996660991, 532092356706983938321, 25283323623228812584415, 1270184310304975912766347 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 becasue 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 REFERENCES Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60.  Solution published in Vol. 39, No. 1, January 2008, pp. 66-67. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..100 Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44.  Mathematical Reviews, MR2433713 (2009c:65129), March 2009.  Zentralblatt MATH, Zbl 1160.65015. A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44. 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} C(n,i)*(i-1)^(i-1)*(n-i)^(n-i) = Sum_{i=1,...,n} C(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 MATHEMATICA Table[Sum[Binomial[n, i] (n - 1)^(n - i), {i, 1, n}], {n, 1, 20}] PROG (MAGMA) [n^n-(n-1)^n: n in [1..20] ]; // Vincenzo Librandi, May 07 2011 (PARI) a(n)=n^n-(n-1)^n; \\ Charles R Greathouse IV, May 08, 2011 (Maxima) a(n) = sum(k!*binomial(n-1, k-1)*stirling2(n, k), k, 1, n); /* Vladimir Kruchinin, Mar 01 2014 */ CROSSREFS Column |a(n, 2)| of A039621. Row sums of triangle A055858. Cf. A000312, A066274, A066275, A047969. Column k=1 of A246049. Sequence in context: A275283 A083071 A305459 * A129481 A276371 A156131 Adjacent sequences:  A045528 A045529 A045530 * A045532 A045533 A045534 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)