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Revision History for A000312

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A000312

a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).
(history; published version)

#323 by Andrey Zabolotskiy at Fri Mar 22 08:19:01 EDT 2024

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#322 by Andrey Zabolotskiy at Fri Mar 22 08:18:58 EDT 2024

FORMULA

a(n) = (n-1)^(n-1)*(2*n) + Sum_{i=1..n-2} binomial(n, i)*(i^i*(n-i-1)^(n-i-1))), , n > 1, a(0) = 1, a(1) = 1. - Vladimir Kruchinin, Nov 28 2014

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#321 by Alois P. Heinz at Tue Feb 13 11:41:50 EST 2024

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#320 by Joerg Arndt at Tue Feb 13 11:40:59 EST 2024

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#319 by Michel Marcus at Tue Feb 13 11:24:42 EST 2024

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#318 by Michel Marcus at Tue Feb 13 11:23:24 EST 2024

LINKS

Nick Hobson, <a href="https://web.archive.org/web/20160413232742/http://www.qbyte.org/puzzles/p048s.html">Solution to puzzle 48: Exponential equation</a>.

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#317 by Alois P. Heinz at Tue Feb 13 10:48:22 EST 2024

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#316 by Michel Marcus at Tue Feb 13 10:34:12 EST 2024

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#315 by Michel Marcus at Tue Feb 13 09:29:00 EST 2024

COMMENTS

All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n = 1, 2, 3, ... . - _Nick Hobson, _, Nov 30 2006

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#314 by Michael De Vlieger at Sun Dec 17 11:21:42 EST 2023

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