A181134 - OEIS

%I #56 Nov 24 2024 09:39:38

%S 0,1,8193,1602516,68711380,1289414505,14350108521,111239118928,

%T 660994932816,3202860761145,13202860761145,47725572905076,

%U 154718778284148,457593884876401,1251308658130545,3197503726489920

%N Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

%C This form of recurrence is a general property of the array in A103438 (sums of the first n-th powers).

%H Bruno Berselli, <a href="/A181134/b181134.txt">Table of n, a(n) for n = 0..10000</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), 2008.

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

%F For n>0, a(n) = n*A123094(n) - Sum_{i=0..n-1} A123094(i), where Sum_{i=0..n-1} A123094(i) = A253712(n-1) = (n-1)*n^2*(n+1)*(30*n^10 - 425*n^8 + 2578*n^6 - 8147*n^4 + 12874*n^2 - 7601)/5460.

%F a(n) = a(-n-1) = (n*(n + 1))^2*(30*n^10 + 150*n^9 + 125*n^8 - 400*n^7 - 326*n^6 + 1052*n^5 + 367*n^4 - 1786*n^3 + 202*n^2 + 1382*n - 691)/420.

%F G.f.: see comment of Vladeta Jovovic in A000538.

%F a(n) = -Sum_{j=1..13} j*Stirling1(n+1,n+1-j)*Stirling2(n+13-j,n). - _Mircea Merca_, Jan 25 2014

%p A181134 := proc(n) (bernoulli(14,n+1) - bernoulli(14))/14 ; end proc: seq(A181134(n), n=0..10); # _R. J. Mathar_, Oct 14 2010

%t Accumulate[Range[0,20]^13] (* _Harvey P. Dale_, Oct 30 2017 *)

%o (Python)

%o A181134_list, m = [0], [6227020800, -37362124800, 97037740800, -142702560000, 130456085760, -76592355840, 28805736960, -6711344640, 901020120, -60780720, 1569750, -8190, 1, 0 , 0]

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

%o     for i in range(14):

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

%o     A181134_list.append(m[-1]) # _Chai Wah Wu_, Nov 06 2014

%o (Magma) [(&+[j^13: j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jul 21 2021

%o (Sage) [(bernoulli_polynomial(n+1, 14) - bernoulli(14))/14  for n in (0..30)] # _G. C. Greubel_, Jul 21 2021

%Y Cf. A010801.

%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), A123094 (m=12), A181134 (m=13).

%Y Cf. A215083, A253712.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Oct 05 2010 - Oct 18 2010