cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A007778 a(n) = n^(n+1).

Original entry on oeis.org

0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.

Links

Crossrefs

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]

Programs

Formula

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

A209060 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz.

Original entry on oeis.org

1, 2, 1, 4, 8, 3, 7, 9, 9, 6, 0, 1, 7, 1, 6, 2, 7, 0, 0, 6, 9, 1, 1, 2, 0, 5, 2, 4, 8, 0, 2, 4, 2, 1, 2, 2, 2, 2, 3, 8, 2, 7, 3, 8, 8, 4, 9, 0, 5, 5, 6, 1, 1, 9, 9, 9, 4, 6, 1, 4, 2, 2, 9, 5, 2, 1, 1, 1, 4, 1, 3, 7, 5, 2, 4, 0, 0, 3, 7, 7, 1, 0, 5, 9, 1, 2, 1, 2, 4, 0, 0, 7, 7, 8, 8, 7, 4, 2, 1, 8, 3, 8, 1
Offset: 1

Views

Author

Peter Bala, Mar 04 2012

Keywords

Comments

Cf. A209059. The double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy equals Integral_{x = 0..1} 1/x^x dx, which is listed as A073009.

Examples

			1.21483799601716270069...

Crossrefs

Cf. A073009, A083648, A135608, A209059.

Programs

  • Mathematica
    digits = 103; 1/2*NSum[ (1/n^n + 1/n^(n+1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 100] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013, from formula *)
  • PARI
    default( realprecision, 105); v = Vec( Str( suminf( n=1, n^-n + n^-(n+1)) / 20)); for( n=3, 105, print1( v[n],",")); /* Michael Somos, Mar 07 2012 */

Formula

The triple integral is most conveniently estimated from the identity Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz = 1/2*Sum_{n = 1..oo} (1/n^n + 1/n^(n+1)).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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