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%I M5460 #53 Sep 08 2022 08:44:35
%S 0,1,513,20196,282340,2235465,12313161,52666768,186884496,574304985,
%T 1574304985,3932252676,9092033028,19696532401,40357579185,78800938560,
%U 147520415296,266108291793,464467582161,787155279940,1299155279940,2093435326521,3300704544313
%N Sum of 9th powers.
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007487/b007487.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H B. Berselli, A description of the recursive method in Formula lines (second formula): website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerSum.html">Power Sum</a>.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F a(n) = n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 (see MathWorld, Power Sum, formula 39). a(n) = n*A000542(n) - Sum_{i=0..n-1} A000542(i). - _Bruno Berselli_, Apr 26 2010
%F G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^11. a(n) = a(-n-1). - _Bruno Berselli_, Aug 23 2011
%F a(n) = -Sum_{j=1..9} j*s(n+1,n+1-j)*S(n+9-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
%F a(n) = (16/5)*A000217(n)^5 - 4*A000217(n)^4 + (12/5)*A000217(n)^3 - (3/5)*A000217(n)^2. - _Michael Raney_, Mar 14 2016
%F a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10. - _Wesley Ivan Hurt_, Dec 21 2016
%p [seq(add(i^9,i=1..n),n=0..40)];
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^9 od: seq(a[n], n=0..22); # _Zerinvary Lajos_, Feb 22 2008
%t lst={};s=0;Do[s=s+n^9;AppendTo[lst, s], {n, 10^2}];lst..or..Table[Sum[k^9, {k, 1, n}], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2008 *)
%t Accumulate[Range[0,30]^9] (* _Harvey P. Dale_, Oct 09 2016 *)
%o (Magma) [&+[n^9: n in [0..m]]: m in [0..22]]; // _Bruno Berselli_, Aug 23 2011
%o (Python)
%o A007487_list, m = [0], [362880, -1451520, 2328480, -1905120, 834120, -186480, 18150, -510, 1, 0, 0]
%o for _ in range(10**2):
%o ....for i in range(10):
%o ........m[i+1]+= m[i]
%o ....A007487_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014
%o (PARI) a(n)=n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Row 9 of array A103438.
%Y Cf. A000217, A000542.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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