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A007778
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a(n) = n^(n+1).
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59
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0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632
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OFFSET
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0,3
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COMMENTS
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Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - Roberto E. Martinez II, Jan 07 2002
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. - Nick Hobson, Nov 30 2006
a(n) is also the number of ways of writing an n-cycle as the product of n+1 transpositions. - Nikos Apostolakis, Nov 22 2008
a(n) is the total number of elements whose preimage is the empty set summed over all partial functions from [n] into [n]. - Geoffrey Critzer, Jan 12 2022
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REFERENCES
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Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nick Hobson, Exponential equation.
Yidong Sun and Jujuan Zhuang, lambda-factorials of n, arXiv:1007.1339 [math.CO], 2010. - Peter Luschny, Jul 09 2010
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FORMULA
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E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - Peter Luschny, Jul 09 2010
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d/dx {x/(T(x)*(1-T(x))}, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012
a(n) = n*A000312(n). - R. J. Mathar, Jan 12 2017
Sum_{n>=2} 1/a(n) = A135608. - Amiram Eldar, Nov 17 2020
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MAPLE
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seq( n^(n+1), n=0..20); # G. C. Greubel, Mar 05 2020
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MATHEMATICA
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Table[n^(n+1), {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)
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PROG
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(MAGMA) [n^(n+1):n in [0..20]]; // Vincenzo Librandi, Jan 03 2012
(Maxima) A007778[n]:=n^(n+1)$
makelist(A007778[n], n, 0, 30); /* Martin Ettl, Oct 29 2012 */
(PARI) vector(21, n, my(m=n-1); m^(m+1)) \\ G. C. Greubel, Mar 05 2020
(Sage) [n^(n+1) for n in (0..20)] # G. C. Greubel, Mar 05 2020
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CROSSREFS
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Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.
Essentially the same as A065440.
Cf. A061250, A143857. [From Reinhard Zumkeller, Jul 23 2010]
Sequence in context: A068617 A207994 A210127 * A065440 A338694 A318047
Adjacent sequences: A007775 A007776 A007777 * A007779 A007780 A007781
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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