A000542 Sum of 8th powers: 1^8 + 2^8 + ... + n^8. (Formerly M5427 N2358)

0, 1, 257, 6818, 72354, 462979, 2142595, 7907396, 24684612, 67731333, 167731333, 382090214, 812071910, 1627802631, 3103591687, 5666482312, 9961449608, 16937207049, 27957167625, 44940730666, 70540730666, 108363590027, 163239463563, 241550448844

Offset

0,3

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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = n*(n+1)*(2*n+1)*(5*n^6 + 15*n^5 + 5*n^4 - 15*n^3 - n^2 + 9*n - 3)/90.

a(n) = n*A000541(n) - Sum_{i=0..n-1} A000541(i). - Bruno Berselli, Apr 26 2010

G.f.: x*(x+1)*(x^6 + 246*x^5 + 4047*x^4 + 11572*x^3 + 4047*x^2 + 246*x + 1)/(x-1)^10. - Colin Barker, May 27 2012

a(n) = 9*a(n-1) - 36* a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) + 40320. - Ant King, Sep 24 2013

a(n) = -Sum_{j=1..8} j*s(n+1,n+1-j)*S(n+8-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

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^8 od: seq(a[n], n=0..23); # Zerinvary Lajos, Feb 22 2008

Mathematica

lst={}; s=0; Do[s=s+n^8; AppendTo[lst, s], {n, 10^2}]; lst..or..Table[Sum[k^8, {k, 1, n}], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
s = 0; lst = {s}; Do[s += n^8; AppendTo[lst, s], {n, 1, 30, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
Accumulate[Range[0, 30]^8] (* Harvey P. Dale, Jun 17 2015 *)

Prog

(Sage) [bernoulli_polynomial(n, 9)/9 for n in range(1, 25)] # Zerinvary Lajos, May 17 2009
(Python)
A000542_list, m = [0], [40320, -141120, 191520, -126000, 40824, -5796, 254, -1, 0, 0]
for _ in range(24):
    for i in range(9):
        m[i+1] += m[i]
    A000542_list.append(m[-1])
print(A000542_list) # Chai Wah Wu, Nov 05 2014
(PARI) a(n)=n*(n+1)*(2*n+1)*(5*n^6+15*n^5+5*n^4-15*n^3-n^2+9*n-3)/90 \\ Charles R Greathouse IV, Sep 28 2015

Crossrefs

Row 8 of array A103438.

Sequence in context: A013956 A294303 A036086 * A023877 A301552 A279641

Adjacent sequences: A000539 A000540 A000541 * A000543 A000544 A000545

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved