%I #57 Oct 15 2024 19:49:47

%S 0,1,2049,179196,4373500,53201625,415998681,2393325424,10983260016,

%T 42364319625,142364319625,427675990236,1170684360924,2962844754961,

%U 7012409924625,15662165784000,33254351828416,67526248136049,131794658215281,248284917113500,453084917113500

%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 Bruno 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*Stirling1(n+1,n+1-j)*Stirling2(n+11-j,n). - _Mircea Merca_, Jan 25 2014

%F a(n) = 1728*A006542(n+2)^2 + 216*A288876(n-2) + 96*A006542(n+2) + A000537(n). - _Yasser Arath Chavez Reyes_, May 25 2024

%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