|
%I M5427 N2358 #65 Jun 28 2023 20:25:37
%S 0,1,257,6818,72354,462979,2142595,7907396,24684612,67731333,
%T 167731333,382090214,812071910,1627802631,3103591687,5666482312,
%U 9961449608,16937207049,27957167625,44940730666,70540730666,108363590027,163239463563,241550448844
%N Sum of 8th powers: 1^8 + 2^8 + ... + n^8.
%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. 815.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%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="/A000542/b000542.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 recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = n*(n+1)*(2*n+1)*(5*n^6 + 15*n^5 + 5*n^4 - 15*n^3 - n^2 + 9*n - 3)/90.
%F a(n) = n*A000541(n) - Sum_{i=0..n-1} A000541(i). - _Bruno Berselli_, Apr 26 2010
%F G.f.: x*(x+1)*(x^6 + 246*x^5 + 4047*x^4 + 11572*x^3 + 4047*x^2 + 246*x + 1)/(x-1)^10. - _Colin Barker_, May 27 2012
%F a(n) = 9*a(n-1) - 36* a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) + 40320. - _Ant King_, Sep 24 2013
%F a(n) = -Sum_{j=1..8} j*s(n+1,n+1-j)*S(n+8-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
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^8 od: seq(a[n], n=0..23); # _Zerinvary Lajos_, Feb 22 2008
%t lst={};s=0;Do[s=s+n^8;AppendTo[lst, s], {n, 10^2}];lst..or..Table[Sum[k^8, {k, 1, n}], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2008 *)
%t s = 0; lst = {s}; Do[s += n^8; AppendTo[lst, s], {n, 1, 30, 1}]; lst (* _Zerinvary Lajos_, Jul 12 2009 *)
%t Accumulate[Range[0,30]^8] (* _Harvey P. Dale_, Jun 17 2015 *)
%o (Sage) [bernoulli_polynomial(n,9)/9 for n in range(1, 25)] # _Zerinvary Lajos_, May 17 2009
%o (Python)
%o A000542_list, m = [0], [40320, -141120, 191520, -126000, 40824, -5796, 254, -1, 0, 0]
%o for _ in range(24):
%o for i in range(9):
%o m[i+1] += m[i]
%o A000542_list.append(m[-1])
%o print(A000542_list) # _Chai Wah Wu_, Nov 05 2014
%o (PARI) a(n)=n*(n+1)*(2*n+1)*(5*n^6+15*n^5+5*n^4-15*n^3-n^2+9*n-3)/90 \\ _Charles R Greathouse IV_, Sep 28 2015
%Y Row 8 of array A103438.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
|
