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%I #46 Sep 08 2022 08:45:28
%S 0,1,2049,179196,4373500,53201625,415998681,2393325424,10983260016,
%T 42364319625,142364319625,427675990236,1170684360924,2962844754961,
%U 7012409924625,15662165784000,33254351828416,67526248136049
%N Sum of first n 11th powers.
%H Vincenzo Librandi, <a href="/A123095/b123095.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F a(n) = n*A023002(n) - Sum_{i=0..n-1} A023002(i). - _Bruno Berselli_, Apr 27 2010
%F a(n) = n^2*(n+1)^2*(2*n^8 +8*n^7 +4*n^6 -16*n^5 -5*n^4 +26*n^3 -3*n^2 -20*n +10)/24. - _Bruno Berselli_, Oct 03 2010
%F G.f.: x*(x^10 +2036*x^9 +152637*x^8 +2203488*x^7 +9738114*x^6 +15724248*x^5 +9738114*x^4 +2203488*x^3 +152637*x^2 +2036*x +1)/(1-x)^13. - _Colin Barker_, May 27 2012
%F a(n) = (-1)*Sum_{j=1..11} j*s(n+1,n+1-j)*S(n+11-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
%p [seq(add(i^11, i=1..n), n=0..20)];
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^11 od: seq(a[n], n=0..13); # _Zerinvary Lajos_, Feb 22 2008
%t Table[Sum[k^11, {k, n}], {n, 0, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2008 *)
%t Accumulate[Range[0,20]^11] (* _Harvey P. Dale_, Sep 17 2021 *)
%o (Python)
%o A123095_list, m = [0], [39916800, -199584000, 419126400, -479001600, 322494480, -129230640, 29607600, -3498000, 171006, -2046, 1, 0 , 0]
%o for _ in range(10**2):
%o ....for i in range(12):
%o ........m[i+1]+= m[i]
%o ....A123095_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014
%o (Magma) [(&+[j^11: j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jul 21 2021
%o (Sage) [(bernoulli_polynomial(n+1, 12) - bernoulli(12))/12 for n in (0..30)] # _G. C. Greubel_, Jul 21 2021
%Y 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), A023002 (m=10), this sequence (m=11), A123094 (m=12), A181134 (m=13).
%Y Cf. A008455, A215083.
%Y Cf. A008275, A008277.
%K nonn,easy
%O 0,3
%A _Zerinvary Lajos_, Sep 27 2006
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