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 A152170 a(n) is the total size of all the image sets of all functions from [n] to [n]. I.e., a(n) is the sum of the cardinalities of every image set of every function whose domain and co-domain is {1,2,...,n}. 2
 0, 1, 6, 57, 700, 10505, 186186, 3805249, 88099320, 2278824849, 65132155990, 2038428376721, 69332064858420, 2546464715771353, 100444826158022178, 4234886922345707265, 190053371487946575856, 9045570064018726951457, 455099825218118626519470 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n)/n^n is the expected value for the cardinality of the image set of a function that takes [n] to [n]. a(n)/(n^(n+1)) is the probability that any particular element of [n] will be in the range of a function f:[n]to[n]. lim_{n->inf} a(n)/(n^(n+1)) = (e-1)/e. - Thomas Dybdahl Ahle, Apr 24 2011 LINKS FORMULA a(n) = n*(n^n - (n-1)^n). a(n) = Sum_{i=1..n} S(n,i)*i!*binomial(n,i)*i where S(n,i) is the Stirling number of the second kind. a(n) = Sum_{k=1..n} A090657(n,k)*k. EXAMPLE a(2) = 6 because the image sets of the functions from  to  are {1},{2},{1,2},{1,2}. MATHEMATICA Table[Sum[StirlingS2[n, i] i! Binomial[n, i] i, {i, 1, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *) PROG (MAGMA) [n*(n^n-(n-1)^n): n in [0..20]]; // Vincenzo Librandi, Jul 23 2017 CROSSREFS Sequence in context: A153851 A141372 A306030 * A087659 A107718 A308863 Adjacent sequences:  A152167 A152168 A152169 * A152171 A152172 A152173 KEYWORD nonn AUTHOR Geoffrey Critzer, Nov 27 2008 EXTENSIONS Added more terms Geoffrey Critzer, Mar 17 2009 STATUS approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)