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A000539 Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.
(Formerly M5241 N2280)
50
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 a(n) = n*A000538(n) - sum(A000538(i), i=0..n-1). - Bruno Berselli, Apr 26 2010

See comment in A008292 for a formula for r-th successive summation of sum(k^j, k=1..n). - Gary Detlefs, Jan 02 2014

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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

a(n) = 6*a(n-1) - 15* a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 120. - Ant King, Sep 23 2013

a(n) = 120*C(n+3,6)+30*C(n+2,4)+C(n+1,2) - Knuth. - Gary Detlefs, Jan 02 2014

a(n)=-sum(j=1..5, j*s(n+1,n+1-j)*S(n+5-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

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, Feb 22 2008

a:=n->sum(j^5, j=0..n): seq(a(n), n=0..30); - Zerinvary Lajos, 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

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 19 17:18 EDT 2014. Contains 240762 sequences.