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%I M5241 N2280 #124 Sep 08 2022 08:44:28
%S 0,1,33,276,1300,4425,12201,29008,61776,120825,220825,381876,630708,
%T 1002001,1539825,2299200,3347776,4767633,6657201,9133300,12333300,
%U 16417401,21571033,28007376,35970000,45735625,57617001,71965908,89176276,109687425,133987425
%N Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.
%C This sequence is related to A000538 by a(n) = n*A000538(n) - Sum_{i=0..n-1} A000538(i). - _Bruno Berselli_, Apr 26 2010
%C See comment in A008292 for a formula for r-th successive summation of Sum_{k=1..n} k^j. - _Gary Detlefs_, Jan 02 2014
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991, p. 275.
%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 T. D. Noe, <a href="/A000539/b000539.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H B. Berselli, A description of the transform in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%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; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FaulhabersFormula.html">Faulhaber's Formula</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber's_formula">Faulhaber's formula</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = n^2*(n+1)^2*(2*n^2+2*n-1)/12.
%F a(n) = sqrt(Sum_{j=1..n}Sum_{i=1..n}(i*j)^5). - _Alexander Adamchuk_, Oct 26 2004
%F a(n) = Sum_{i = 1..n} J_5(i)*floor(n/i), where J_5 is A059378. - _Enrique Pérez Herrero_, Feb 26 2012
%F a(n) = 6*a(n-1) - 15* a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 120. - _Ant King_, Sep 23 2013
%F a(n) = 120*C(n+3,6) + 30*C(n+2,4) + C(n+1,2) (Knuth). - _Gary Detlefs_, Jan 02 2014
%F a(n) = -Sum_{j=1..5} j*s(n+1,n+1-j)*S(n+5-j,n)), where s(n,k) and S(n,k) are the Stirling numbers of the first kind and the second kind, respectively. - _Mircea Merca_, Jan 25 2014
%F Sum_{n>=1} 1/a(n) = 60 - 4*Pi^2 + 8*sqrt(3)*Pi * tan(sqrt(3)*Pi/2). - _Vaclav Kotesovec_, Feb 13 2015
%F a(n) = (n + 1)^2*n^2*(n + 1/2 + sqrt(3/4))*(n + 1/2 - sqrt(3/4))/6. See the Graham et al. reference, p. 275. - _Wolfdieter Lang_, Apr 02 2015
%F G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)^7. - _Robert Israel_, Dec 07 2015
%F a(n) = (4/3)*A000217(n)^3 - (1/3)*A000217(n)^2. - _Michael Raney_, Feb 19 2016
%F a(n) = (binomial(n+1,4) + 6*binomial(n+2,4) + binomial(n+3,4))*(binomial(n+2,3) - binomial(n+1,3)). - _Tony Foster III_, Oct 21 2018
%p A000539:=-(1+26*z+66*z**2+26*z**3+z**4)/(z-1)**7; # _Simon Plouffe_ in his 1992 dissertation
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^5 od: seq(a[n], n=0..30); # _Zerinvary Lajos_, Feb 22 2008
%p a:=n->sum(j^5,j=0..n): seq(a(n), n=0..30); # _Zerinvary Lajos_, Jun 05 2008
%t Accumulate[Range[0, 40]^5]
%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 33, 276, 1300, 4425, 12201}, 41] (* _Jean-François Alcover_, Feb 09 2016 *)
%o (PARI) a(n)=n^2*(n+1)^2*(2*n^2+2*n-1)/12 \\ _Charles R Greathouse IV_, Jul 15 2011
%o (Maxima) A000539(n):=n^2*(n+1)^2*(2*n^2+2*n-1)/12$ makelist(A000539(n),n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [n^2*(n+1)^2*(2*n^2+2*n-1)/12: n in [0..30]]; // _Vincenzo Librandi_, Apr 04 2015
%o (Python)
%o A000539_list, m = [0], [120, -240, 150, -30, 1, 0, 0]
%o for _ in range(10**2):
%o for i in range(6):
%o m[i+1] += m[i]
%o A000539_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014
%o (PARI) concat(0, Vec(x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)^7 + O(x^100))) \\ _Altug Alkan_, Dec 07 2015
%Y Partial sums of A000584. Row 5 of array A103438.
%Y Cf. A000538, A101092.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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