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A152170
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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}.
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0
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0, 1, 6, 57, 700, 10505, 186186, 3805249, 88099320, 2278824849, 65132155990, 2038428376721, 69332064858420, 2546464715771353, 100444826158022178, 4234886922345707265, 190053371487946575856, 9045570064018726951457
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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].
a(n)/(n^(n+1)) goes to (e-1)/e as n goes to infinity. [Thomas Dybdahl Ahle, Apr 24 2011]
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FORMULA
| a(n)=n(n^n-(n-1)^n); a(n) = Sum_{i=1..n} S(n,i)*i!*C(n,i)*i where S(n,i) is the Stirling number of the second kind and C(n,i) is the binomial coefficient.
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EXAMPLE
| a(2)=6 because the image sets of the functions from [2] to [2] are {1},{2},{1,2},{1,2}
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MATHEMATICA
| Table[Sum[StirlingS2[n, i] i! Binomial[n, i] i, {i, 1, n}], {n, 0, 20}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 17 2009]
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CROSSREFS
| Sequence in context: A060435 A153851 A141372 * A087659 A107718 A000406
Adjacent sequences: A152167 A152168 A152169 * A152171 A152172 A152173
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KEYWORD
| nonn
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AUTHOR
| Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 27 2008
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EXTENSIONS
| Added more terms Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 17 2009
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