login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000541 Sum of 7th powers: 1^7 + 2^7 + ... + n^7.
(Formerly M5394 N2343)
15
0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456, 8080425, 18080425, 37567596, 73399404, 136147921, 241561425, 412420800, 680856256, 1091194929, 1703414961, 2597286700, 3877286700, 5678375241, 8172733129, 11577558576, 16164030000, 22267545625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is divisible by A000537(n) if and only n is congruent to 1 mod 3 (see A016777) - Artur Jasinski, Oct 10 2007

This sequence is related to A000540 by a(n) = n*A000540(n)-sum(A000540(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. 815.

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 recursive method in Comments lines: website Matem@ticamente (in Italian).

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24.

a(n) = Sqrt[Sum[Sum[(i*j)^7, {i, 1, n}], {j, 1, n}]]. - Alexander Adamchuk, Oct 26 2004

Jacobi formula: a(n) = 2(A000217(n))^4 - A000539(n). - Artur Jasinski, Oct 10 2007

G.f.: x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/(1-x)^9. - Colin Barker, May 25 2012

a(n) = 8*a(n-1) - 28* a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 5040. - Ant King, Sep 24 2013

a(n) = -sum(j=1..7, j*s(n+1,n+1-j)*S(n+7-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

a(n) = 2*A000217(n)^4 - (4/3)*A000217(n)^3 + (1/3)*A000217(n)^2. - Michael Raney, Feb 19 2016

MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^7 od: seq(a[n], n=0..25); # Zerinvary Lajos, Feb 22 2008

MATHEMATICA

Table[Sum[k^7, {k, 1, n}], {n, 0, 100}] (* Artur Jasinski, Oct 10 2007 *)

s = 0; lst = {s}; Do[s += n^7; AppendTo[lst, s], {n, 1, 30, 1}]; lst (* Zerinvary Lajos, Jul 12 2009

LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456}, 35] (* Vincenzo Librandi, Feb 20 2016 *)

PROG

(PARI) a(n)=n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24 \\ Edward Jiang, Sep 10 2014

(PARI) a(n) = sum(i=1, n, i^7); \\ Michel Marcus, Sep 11 2014

(Python)

A000541_list, m = [0], [5040, -15120, 16800, -8400, 1806, -126, 1, 0, 0]

for _ in range(10**2):

....for i in range(8):

........m[i+1] += m[i]

....A000541_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

(MAGMA) [n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24: n in [0..30]]; // Vincenzo Librandi, Feb 20 2016

CROSSREFS

Row 7 of array A103438.

Cf. A000217, A000537, A000539.

Sequence in context: A294302 A221969 A036085 * A023876 A301551 A297493

Adjacent sequences:  A000538 A000539 A000540 * A000542 A000543 A000544

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 28 03:59 EDT 2018. Contains 304733 sequences. (Running on oeis4.)