A000540 Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6. (Formerly M5335 N2322)

0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, 1978405, 3749966, 6735950, 11562759, 19092295, 30482920, 47260136, 71397705, 105409929, 152455810, 216455810, 302221931, 415601835, 563637724, 754740700, 998881325, 1307797101, 1695217590

Offset

0,3

Comments

This sequence is related to A000539 by a(n) = n*A000539(n)-sum(A000539(i), i=0..n-1). - Bruno Berselli, Apr 26 2010

References

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.

J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, (2008), p. 289.

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

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]

B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1)

Formula

a(n) = n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42.

a(n) = sqrt(Sum_{j=1..n} Sum_{i=1..n} (i*j)^6). - Alexander Adamchuk, Oct 26 2004

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

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

a(n) = Sum_{i=1..n} J_6(i)*floor(n/i), where J_6 is A069091. - Enrique Pérez Herrero, Mar 09 2013

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

a(n) = -Sum_{j=1..6} j*s(n+1,n+1-j)*S(n+6-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

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

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

Maple

a:=n->sum (j^6, j=0..n): seq(a(n), n=0..27); # Zerinvary Lajos, Jun 27 2007
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.
A000540 := proc(n) n^7/7+n^6/2+n^5/2-n^3/6+n/42 ; end proc: # R. J. Mathar

Mathematica

Accumulate[Range[0, 30]^6] (* Harvey P. Dale, Jul 30 2009 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 1, 65, 794, 4890, 20515, 67171, 184820}, 31] (* Jean-François Alcover, Feb 09 2016 *)

Prog

(Sage) [bernoulli_polynomial(n, 7)/7 for n in range(1, 29)]# Zerinvary Lajos, May 17 2009
(Haskell)
a000540 n = a000540_list !! n
a000540_list = scanl1 (+) a001014_list -- Reinhard Zumkeller, Dec 04 2011
(PARI) a(n)=n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42 \\ Edward Jiang, Sep 10 2014
(PARI) a(n)=sum(i=1, n, i^6); \\ Michel Marcus, Sep 11 2014
(Python)
A000540_list, m = [0], [720, -1800, 1560, -540, 62, -1, 0, 0]
for _ in range(10**2):
    for i in range(7):
        m[i+1] += m[i]
    A000540_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014
(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

Crossrefs

Cf. A101093, A000539.

Row 6 of array A103438.

Partial sums of A001014.

Sequence in context: A196634 A196639 A008516 * A023875 A301550 A027463

Adjacent sequences: A000537 A000538 A000539 * A000541 A000542 A000543

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved