|
| |
|
|
A000539
|
|
Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.
(Formerly M5241 N2280)
|
|
48
|
|
|
|
0, 1, 33, 276, 1300, 4425, 12201, 29008, 61776, 120825, 220825, 381876, 630708, 1002001, 1539825, 2299200, 3347776, 4767633, 6657201, 9133300, 12333300, 16417401, 21571033, 28007376, 35970000, 45735625, 57617001, 71965908, 89176276, 109687425, 133987425
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
This sequence is related to A000538 by the transform a(n) = n*A000538(n) - sum(A000538(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.
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).
|
|
|
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 transform in Comments lines: website Matem@ticamente (in Italian).
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein, MathWorld: Faulhaber's Formula
Wikipedia, Faulhaber's formula
Index to sequences with linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
|
|
|
FORMULA
|
a(n) = n^2*(n+1)^2*(2*n^2+2*n-1)/12.
a(n) = Sqrt[Sum[Sum[(i*j)^5, {i, 1, n}], {j, 1, n}]]. - Alexander Adamchuk, Oct 26 2004
a(n) = sum (i = 1 .. n, J_ 5 (i)*floor (n/i)), where J_ 5 is A059378. - Enrique Pérez Herrero, Feb 26 2012
|
|
|
MAPLE
|
A000539:=-(1+26*z+66*z**2+26*z**3+z**4)/(z-1)**7; [Simon Plouffe in his 1992 dissertation.]
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 (zerinvarylajos(AT)yahoo.com), Feb 22 2008
a:=n->sum(j^5, j=0..n): seq(a(n), n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2008
|
|
|
MATHEMATICA
|
Accumulate[Range[0, 40]^5]
|
|
|
PROG
|
(PARI) a(n)=n^2*(n+1)^2*(2*n^2+2*n-1)/12 \\ Charles R Greathouse IV, Jul 15 2011
(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 */
|
|
|
CROSSREFS
|
Partial sums of A000584. Row 5 of array A103438.
Cf. A101092, A000538.
Sequence in context: A119782 A008515 A179995 * A023874 A020291 A222726
Adjacent sequences: A000536 A000537 A000538 * A000540 A000541 A000542
|
|
|
KEYWORD
|
nonn,easy,changed
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
