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A023002 Sum of 10th powers.

%I A023002 #69 Jul 18 2025 08:34:46
%S A023002 0,1,1025,60074,1108650,10874275,71340451,353815700,1427557524,
%T A023002 4914341925,14914341925,40851766526,102769130750,240627622599,
%U A023002 529882277575,1106532668200,2206044295976,4222038196425,7792505423049,13923571680850
%N A023002 Sum of 10th powers.
%H A023002 T. D. Noe, <a href="/A023002/b023002.txt">Table of n, a(n) for n = 0..1000</a>
%H A023002 Bruno 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 A023002 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 13.
%H A023002 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerSum.html">Power Sum</a>.
%H A023002 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A023002 a(n) = n*(n+1)*(2*n+1)*(n^2+n-1)(3*n^6 +9*n^5 +2*n^4 -11*n^3 +3*n^2 +10*n -5)/66 (see MathWorld, Power Sum, formula 40). - _Bruno Berselli_, Apr 26 2010
%F A023002 a(n) = n*A007487(n) - Sum_{i=0..n-1} A007487(i). - _Bruno Berselli_, Apr 27 2010
%F A023002 From _Bruno Berselli_, Aug 23 2011: (Start)
%F A023002 a(n) = -a(-n-1).
%F A023002 G.f.: x*(1+x)*(1 +1012*x +46828*x^2 +408364*x^3 +901990*x^4 +408364*x^5 +46828*x^6 +1012*x^7 +x^8)/(1-x)^12. (End)
%F A023002 a(n) = (-1)*Sum_{j=1..10} j*Stirling1(n+1,n+1-j)*Stirling2(n+10-j,n). - _Mircea Merca_, Jan 25 2014
%F A023002 a(n) = Sum_{i=1..n} J_10(i)*floor(n/i), where J_10 is A069095. - _Ridouane Oudra_, Jul 17 2025
%p A023002 A023002:= n-> bernoulli(11, n+1)/11; seq(A023002(n), n=0..30); # _G. C. Greubel_, Jul 21 2021
%t A023002 Table[Sum[k^10, {k, n}], {n, 0, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2008 *)
%t A023002 Accumulate[Range[0,20]^10] (* _Harvey P. Dale_, Aug 23 2011 *)
%o A023002 (Sage) [bernoulli_polynomial(n,11)/11 for n in range(2, 21)]# _Zerinvary Lajos_, May 17 2009
%o A023002 (Magma) [&+[n^10: n in [0..m]]: m in [0..19]];  // _Bruno Berselli_, Aug 23 2011
%o A023002 (PARI) a(n)=(6*x^11+33*x^10+55*x^9-66*x^7+66*x^5-33*x^3+5*x)/66 \\ _Charles R Greathouse IV_, Aug 23 2011
%o A023002 (PARI) a(n)=sum(i=0,10,binomial(11,i)*bernfrac(i)*n^(11-i))/11+n^10 \\ _Charles R Greathouse IV_, Aug 23 2011
%o A023002 (Python)
%o A023002 A023002_list, m = [0], [3628800, -16329600, 30240000, -29635200, 16435440, -5103000, 818520, -55980, 1022, -1, 0 , 0]
%o A023002 for _ in range(20):
%o A023002     for i in range(11):
%o A023002         m[i+1]+= m[i]
%o A023002     A023002_list.append(m[-1])
%o A023002 print(A023002_list) # _Chai Wah Wu_, Nov 05 2014
%Y A023002 Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), this sequence (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).
%Y A023002 Row 10 of array A103438.
%Y A023002 Cf. A215083, A069095.
%K A023002 nonn,easy
%O A023002 0,3
%A A023002 _David W. Wilson_