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A000540 Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.

%I A000540 M5335 N2322 #135 Jan 06 2025 11:58:26
%S A000540 0,1,65,794,4890,20515,67171,184820,446964,978405,1978405,3749966,
%T A000540 6735950,11562759,19092295,30482920,47260136,71397705,105409929,
%U A000540 152455810,216455810,302221931,415601835,563637724,754740700,998881325,1307797101,1695217590
%N A000540 Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.
%C A000540 This sequence is related to A000539 by a(n) = n*A000539(n)-sum(A000539(i), i=0..n-1). - _Bruno Berselli_, Apr 26 2010
%D A000540 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 A000540 J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
%D A000540 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D A000540 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, (2008), p. 289.
%D A000540 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000540 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000540 T. D. Noe, <a href="/A000540/b000540.txt">Table of n, a(n) for n = 0..1000</a>
%H A000540 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 A000540 J. L. Bailey, <a href="/A002309/a002309.pdf">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
%H A000540 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 A000540 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 13.
%H A000540 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 A000540 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000540 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1)
%F A000540 a(n) = n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42.
%F A000540 a(n) = sqrt(Sum_{j=1..n} Sum_{i=1..n} (i*j)^6). - _Alexander Adamchuk_, Oct 26 2004
%F A000540 G.f.: A(x) = 3*x/7*G(0); with G(k) = 1 + 2/(k+1+(k+1)/(2*k^2 + 4*k + 1 + 2*(k+1)^2/(3*k + 2 - 9*x*(k+1)*(k+2)^4*(k+3)*(2*k+5)/(3*x*(k+2)^4*(k+3)*(2*k+5)+(k+1)*(2*k+3)/G(k+1))))); (continued fraction). - _Sergei N. Gladkovskii_, Dec 03 2011
%F A000540 G.f.: x*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1) / (x-1)^8 . - _R. J. Mathar_, Aug 07 2012
%F A000540 a(n) = Sum_{i=1..n} J_6(i)*floor(n/i), where J_6 is A069091. - _Enrique Pérez Herrero_, Mar 09 2013
%F A000540 a(n) = 7*a(n-1) - 21* a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + 720. - _Ant King_, Sep 24 2013
%F A000540 a(n) = -Sum_{j=1..6} j*Stirling1(n+1,n+1-j)*Stirling2(n+6-j,n). - _Mircea Merca_, Jan 25 2014
%F A000540 Sum_{n>=1} (-1)^(n+1)/a(n) = 84*Pi*(8*cos(sqrt((sqrt(93) + 9)/6)*Pi) + 15*cos(sqrt((sqrt(93) + 9)/6)*Pi/2) * cosh(sqrt((sqrt(93) - 9)/6)*Pi/2) + 8*cosh(sqrt((sqrt(93) - 9)/6)*Pi) - 7*sqrt(3)*sin(sqrt((sqrt(93) + 9)/6)*Pi/2) * sinh(sqrt((sqrt(93) - 9)/6)*Pi/2)) / (31*(cos(sqrt((sqrt(93) + 9)/6)*Pi) + cosh(sqrt((sqrt(93) - 9)/6)*Pi))) = 0.985708051237101247832970793342271511... . - _Vaclav Kotesovec_, Feb 13 2015
%F A000540 a(n) = (n + 1)*(n + 1/2)*n*(n + 1/2 + z)*(n + 1/2 - z)*(n + 1/2 + zbar)*(n + 1/2 - zbar)/7, with I^2 = -1 and z = 2^(-3/2)*3^(-1/4)*(sqrt(sqrt(31) + 3*sqrt(3)) + I*sqrt(sqrt(31) - 3*sqrt(3))), and zbar is the complex conjugate of z. See the Graham et al. reference, eq. (6.98), pp. 288-289 (with n -> n+1). (There was a typo in the first edition, which was corrected in the second edition.) - _Wolfdieter Lang_, Apr 03 2015
%F A000540 a(n+2) = 36*A086020(n+1) + 24*A005585(n+1) + A000330(n+2). - _Yasser Arath Chavez Reyes_, Apr 16 2024
%p A000540 a:=n->sum (j^6,j=0..n): seq(a(n),n=0..27); # _Zerinvary Lajos_, Jun 27 2007
%p A000540 A000540:=(z+1)*(z**4+56*z**3+246*z**2+56*z+1)/(z-1)**8; # g.f. by _Simon Plouffe_ in his 1992 dissertation, without the leading 0.
%p A000540 A000540 := proc(n) n^7/7+n^6/2+n^5/2-n^3/6+n/42 ; end proc: # _R. J. Mathar_
%t A000540 Accumulate[Range[0,30]^6] (* _Harvey P. Dale_, Jul 30 2009 *)
%t A000540 LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 1, 65, 794, 4890, 20515, 67171, 184820}, 31] (* _Jean-François Alcover_, Feb 09 2016 *)
%o A000540 (Sage) [bernoulli_polynomial(n,7)/7 for n in range(1, 29)]# _Zerinvary Lajos_, May 17 2009
%o A000540 (Haskell)
%o A000540 a000540 n = a000540_list !! n
%o A000540 a000540_list = scanl1 (+) a001014_list -- _Reinhard Zumkeller_, Dec 04 2011
%o A000540 (PARI) a(n)=n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42 \\ _Edward Jiang_, Sep 10 2014
%o A000540 (PARI) a(n)=sum(i=1, n, i^6); \\ _Michel Marcus_, Sep 11 2014
%o A000540 (Python)
%o A000540 A000540_list, m = [0], [720, -1800, 1560, -540, 62, -1, 0, 0]
%o A000540 for _ in range(10**2):
%o A000540     for i in range(7):
%o A000540         m[i+1] += m[i]
%o A000540     A000540_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014
%o A000540 (Magma) [n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42: n in [0..30]]; // _Vincenzo Librandi_, Apr 04 2015
%Y A000540 Cf. A101093, A000539.
%Y A000540 Row 6 of array A103438.
%Y A000540 Partial sums of A001014.
%K A000540 nonn,easy
%O A000540 0,3
%A A000540 _N. J. A. Sloane_