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 A254871 #32 Sep 08 2022 08:46:11
%S A254871 1,39,495,3705,19995,85917,311493,989235,2823990,7383610,17931498,
%T A254871 40889862,88304970,181852230,359140470,683363994,1257722271,
%U A254871 2246496825,3905261425,6623425575,10983195405,17840105595,28431558675,44521334325,68589834300,104081944356
%N A254871 Seventh partial sums of fifth powers (A000584).
%H A254871 Luciano Ancora, <a href="/A254871/b254871.txt">Table of n, a(n) for n = 1..1000</a>
%H A254871 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>
%H A254871 Luciano Ancora, <a href="/A254647/a254647_2.pdf">Pascal’s triangle and recurrence relations for partial sums of m-th powers</a>
%H A254871 <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 A254871 G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^13.
%F A254871 a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680.
%F A254871 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^5.
%e A254871 Second differences: 30, 180, 570, 1320, 2550, ... (A068236)
%e A254871 First differences: 1, 31, 211, 781, 2101, 4651, ... (A022521)
%e A254871 ------------------------------------------------------------------------
%e A254871 The fifth powers: 1, 32, 243, 1024, 3125, 7776, ... (A000584)
%e A254871 ------------------------------------------------------------------------
%e A254871 First partial sums: 1, 33, 276, 1300, 4425, 12201, ... (A000539)
%e A254871 Second partial sums: 1, 34, 310, 1610, 6035, 18236, ... (A101092)
%e A254871 Third partial sums: 1, 35, 345, 1955, 7990, 26226, ... (A101099)
%e A254871 Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (A254644)
%e A254871 Fifth partial sums: 1, 37, 418, 2754, 13080, 49632, ... (A254682)
%e A254871 Sixth partial sums: 1, 38, 456, 3210, 16290, 65922, ... (A254471)
%e A254871 Seventh partial sums: 1, 39, 495, 3705, 19995, 85917, ... (this sequence)
%t A254871 Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) ((-21 + 49 n + 56 n^2 + 14 n^3 + n^4)/3991680), {n, 23}] (* or *)
%t A254871 CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^13, {x, 0, 22}], x]
%o A254871 (PARI) vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680) \\ _Derek Orr_, Feb 19 2015
%o A254871 (Magma) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(-21+49*n +56*n^2+14*n^3+n^4)/3991680: n in [1..30]]; // _Vincenzo Librandi_, Feb 19 2015
%Y A254871 Cf. A000539, A000584, A022521, A101092, A101099, A254471, A254644, A254682, A254869, A254870, A254872.
%K A254871 nonn,easy
%O A254871 1,2
%A A254871 _Luciano Ancora_, Feb 17 2015