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A000584 Fifth powers: a(n) = n^5.
(Formerly M5231 N2277)
158

%I M5231 N2277

%S 0,1,32,243,1024,3125,7776,16807,32768,59049,100000,161051,248832,

%T 371293,537824,759375,1048576,1419857,1889568,2476099,3200000,4084101,

%U 5153632,6436343,7962624,9765625,11881376,14348907,17210368,20511149

%N Fifth powers: a(n) = n^5.

%C Totally multiplicative sequence with a(p) = p^5 for prime p. - _Jaroslav Krizek_, Nov 01 2009

%C The binomial transform yields A059338. The inverse binomial transform yields the (finite) 0, 1, 30, 150, 240, 120, the 5th row in A019538 and A131689. - _R. J. Mathar_, Jan 16 2013

%C Equals sum of odd numbers from n^2*(n-1)+1 (A100104) to n^2*(n+1)-1 (A003777). - _Bruno Berselli_, Mar 14 2014

%C a(n) mod 10 = n mod 10. - _Reinhard Zumkeller_, May 10 2014

%C Numbers of the form a(n) + a(n+1) + ... + a(n+k) are nonprime for all n, k>=0; this can be proved by the method indicated in the comment in A256581. - _Vladimir Shevelev_ and _Peter J. C. Moses_, Apr 04 2015

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Franklin T. Adams-Watters, <a href="/A000584/b000584.txt">Table of n, a(n) for n = 0..500</a>

%H Brady Haran and Simon Pampena, <a href="https://www.youtube.com/watch?v=y8acoaakvPM">Fifth Root Trick</a>, Numberphile video (2014)

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (x-1)^6. [_Simon Plouffe_ in his 1992 dissertation]

%F Multiplicative with a(p^e) = p^(5e). - _David W. Wilson_, Aug 01 2001

%F E.g.f.: exp(x)*(x+15*x^2+25*x^3+10*x^4+x^5). - _Geoffrey Critzer_, Jun 12 2013

%F a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 120. - _Ant King_, Sep 23 2013

%F a(n) = n + Sum_{j=0..n-1}{k=1..4}binomial(5,k)*j^(5-k). - _Patrick J. McNab_, Mar 28 2016

%F From _Kolosov Petro_, Oct 22 2018: (Start)

%F a(n) = Sum_{k=1..n} A300656(n,k).

%F a(n) = Sum_{k=0..n-1} A300656(n,k). (End)

%F a(n) = Sum_{k=1..5} Eulerian(5, k)*binomial(n+5-k, 5), with Eulerian(5, k) = A008292(5, k), the numbers 1, 26, 66, 26, 1, for n >= 0. Worpitzki's identity for powers of 5. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - _Wolfdieter Lang_, Jul 17 2019

%F From _Amiram Eldar_, Oct 08 2020: (Start)

%F Sum_{n>=1} 1/a(n) = zeta(5) (A013663).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 15*zeta(5)/16 (A267316). (End)

%t Range[0,50]^5 (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011 *)

%o (Sage) [n**5 for n in range(30)] # _Zerinvary Lajos_, Jun 03 2009

%o (Haskell)

%o a000584 = (^ 5) -- _Reinhard Zumkeller_, Nov 11 2012

%o (Maxima) makelist(n^5, n, 0, 30); /* _Martin Ettl_, Nov 12 2012 */

%o (PARI) a(n)=n^5 \\ _Charles R Greathouse IV_, Jul 28 2015

%Y Partial sums give A000539.

%Y Cf. A000012, A001477, A000290, A000578, A000583, A062392, A022521 (first differences), A008292, A173018, A123125.

%Y Cf. A013663, A162624, A267316.

%K nonn,easy,mult

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Henry Bottomley_, Jun 21 2001

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Last modified July 31 23:46 EDT 2021. Contains 346377 sequences. (Running on oeis4.)