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A000540
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Sum of 6th powers: 1^6 + 2^6 + ... + n^6.
(Formerly M5335 N2322)
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17
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = n*A000539(n) - sum [i = 0 ... n-1] A000539(i) [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 26 2010]
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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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
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).
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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].
B. Berselli, a description of the recursive method n*Ar(n)-sum[i=0...n-1]Ar(i) (Ar(m) is the m-th term of a sequence): website Matem@ticamente. [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 26 2010]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n) = n*(n+1)*(2*n+1)*(3*n^4+6*n^3-3*n+1)/42.
a(n) = Sqrt[Sum[Sum[(i*j)^6, {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), 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
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MAPLE
| a:=n->sum (j^6, j=0..n): seq(a(n), n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007
A000540:=(z+1)*(z**4+56*z**3+246*z**2+56*z+1)/(z-1)**8; [G.f. by S. Plouffe in his 1992 dissertation, without the leading 0.]
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^6 od: seq(a[n], n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2008
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MATHEMATICA
| lst={}; s=0; Do[s=s+n^6; AppendTo[lst, s], {n, 10^2}]; lst..or..Table[Sum[k^6, {k, 1, n}], {n, 0, 100}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 14 2008]
s = 0; lst = {s}; Do[s += n^6; AppendTo[lst, s], {n, 1, 30, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
Accumulate[Range[0, 30]^6] [From Harvey P. Dale (hpd1(AT)nyu.edu), Jul 30 2009]
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PROG
| (Other) sage: [bernoulli_polynomial(n, 7)/7 for n in xrange(1, 29)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(Haskell)
a000540 n = a000540_list !! n
a000540_list = scanl1 (+) a001014_list -- Reinhard Zumkeller, Dec 04 2011
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CROSSREFS
| Cf. A101093.
Row 6 of array A103438.
Partial sums of A001014.
Sequence in context: A196634 A196639 A008516 * A023875 A027463 A144500
Adjacent sequences: A000537 A000538 A000539 * A000541 A000542 A000543
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Significantly simpler and faster Mathematica program provided. [From Harvey P. Dale (hpd1(AT)nyu.edu), Jul 30 2009]
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