A000542 Sum of 8th powers: 1^8 + 2^8 + ... + n^8.
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
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].
- Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Programs
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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
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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 *) -
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
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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
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Sage
[bernoulli_polynomial(n,9)/9 for n in range(1, 25)] # Zerinvary Lajos, May 17 2009
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.
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*Stirling1(n+1,n+1-j)*Stirling2(n+8-j,n). - Mircea Merca, Jan 25 2014
a(n) = Sum_{i = 1..n} J_8(i)*floor(n/i), where J_8 is A069093. - Ridouane Oudra, Jul 17 2025