A123094 - OEIS

%I #46 May 27 2024 12:58:22

%S 0,1,4097,535538,17312754,261453379,2438235715,16279522916,

%T 84998999652,367428536133,1367428536133,4505856912854,13421957361110,

%U 36720042483591,93413954858887,223160292749512,504635269460168,1087257506689929,2244088888116105,4457403807182266

%N Sum of first n 12th powers.

%H T. D. Noe, <a href="/A123094/b123094.txt">Table of n, a(n) for n = 0..1000</a>

%H Bruno Berselli, A description of the recursive method in Formula lines (first formula): website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%F a(n) = n*A123095(n) - Sum_{i=0..n-1} A123095(i). - _Bruno Berselli_, Apr 27 2010

%F a(n) = n * (n+1) * (2*n+1) * (105*n^10 +525*n^9 +525*n^8 -1050*n^7 -1190*n^6 +2310*n^5 +1420*n^4 -3285*n^3 -287*n^2 +2073*n -691)/2730. - _Bruno Berselli_, Oct 03 2010

%F a(n) = (-1)*Sum_{j=1..12} j*Stirling1(n+1,n+1-j)*Stirling2(n+12-j,n). - _Mircea Merca_, Jan 25 2014

%p [seq(add(i^12, i=1..n), n=0..18)];

%t Table[Sum[k^12, {k, n}], {n, 0, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2008 *)

%t Accumulate[Range[0,30]^12]  (* _Harvey P. Dale_, Apr 26 2011 *)

%o (Sage) [bernoulli_polynomial(n,13)/13 for n in range(1, 30)] # _Zerinvary Lajos_, May 17 2009

%o (Python)

%o A123094_list, m = [0], [479001600, -2634508800, 6187104000, -8083152000, 6411968640, -3162075840, 953029440, -165528000, 14676024, -519156, 4094, -1, 0 , 0]

%o for _ in range(10**2):

%o     for i in range(13):

%o         m[i+1]+= m[i]

%o     A123094_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014

%o (Magma) [(&+[j^12: j in [0..n]]): j 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), A123095 (m=11), this sequence (m=12), A181134 (m=13).

%Y Cf. A008456, A215083.

%Y Cf. A008275, A008277.

%K easy,nonn

%O 0,3

%A _Zerinvary Lajos_, Sep 27 2006