This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A254472 #27 Jan 27 2022 03:09:02
%S A254472 1,70,1134,9870,59220,275562,1063530,3552978,10577385,28652260,
%T A254472 71725108,167911380,371057232,779831820,1568210220,3032733564,
%U A254472 5663906745,10251608346,18037546450,30931714450,51814612980,84952851750,136562787270,215565263550,334584493425
%N A254472 Sixth partial sums of sixth powers (A001014).
%H A254472 Luciano Ancora, <a href="/A254472/b254472.txt">Table of n, a(n) for n = 1..1000</a>
%H A254472 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>.
%H A254472 Luciano Ancora, <a href="/A254647/a254647_2.pdf">Pascal’s triangle and recurrence relations for partial sums of m-th powers</a>.
%H A254472 <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 A254472 G.f.: (-x - 57*x^2 - 302*x^3 - 302*x^4 - 57*x^5 - x^6)/(- 1 + x)^13.
%F A254472 a(n) = n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(-3 + 5*n + n^2)*(3 + 7*n + n^2)/665280.
%F A254472 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^6.
%F A254472 Sum_{n>=1} 1/a(n) = 25622179/76545 - 3080*Pi^2/81 + 149600*Pi*tan(sqrt(37)*Pi/2)/(243*sqrt(37)). - _Amiram Eldar_, Jan 27 2022
%e A254472 First differences: 1, 63, 665, 3367, 11529, ... (A022522)
%e A254472 --------------------------------------------------------------------------
%e A254472 The sixth powers: 1, 64, 729, 4096, 15625, ... (A001014)
%e A254472 --------------------------------------------------------------------------
%e A254472 First partial sums: 1, 65, 794, 4890, 20515, ... (A000540)
%e A254472 Second partial sums: 1, 66, 860, 5750, 26265, ... (A101093)
%e A254472 Third partial sums: 1, 67, 927, 6677, 32942, ... (A254640)
%e A254472 Fourth partial sums: 1, 68, 995, 7672, 40614, ... (A254645)
%e A254472 Fifth partial sums: 1, 69, 1064, 8736, 49350, ... (A254683)
%e A254472 Sixth partial sums: 1, 70, 1134, 9870, 59220, ... (this sequence)
%t A254472 Table[n (1 + n) (2 + n) (3 + n)^2 (4 + n) (5 + n) (6 + n) (- 3 + 5 n + n^2) (3 + 7 n + n^2)/665280, {n, 22}] (* or *) CoefficientList[Series[(- 1 - 57 x - 302 x^2 - 302 x^3 - 57 x^4 - x^5)/(- 1 + x)^13, {x, 0, 28}], x]
%t A254472 Nest[Accumulate,Range[30]^6,6] (* _Harvey P. Dale_, Oct 02 2015 *)
%o A254472 (Magma) [n*(1+n)*(2+n)*(3+n)^2*(4+n)*(5+n)*(6+n)*(-3+5*n+n^2)* (3+7*n+n^2)/665280: n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2015
%o A254472 (PARI) vector(50,n,n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(-3 + 5*n + n^2)*(3 + 7*n + n^2)/665280) \\ _Derek Orr_, Feb 19 2015
%Y A254472 Cf. A000540, A001014, A022522, A101093, A254469, A254470, A254471, A254640, A254645, A254683.
%K A254472 nonn,easy
%O A254472 1,2
%A A254472 _Luciano Ancora_, Feb 15 2015