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 A254469 #47 Jan 28 2025 10:52:29
%S A254469 1,14,96,450,1650,5082,13728,33462,75075,157300,311168,586092,1058148,
%T A254469 1841100,3100800,5073684,8090181,12603954,19228000,28778750,42329430,
%U A254469 61274070,87403680,122996250,170922375,234768456,318979584,429024376,571584200,754769400
%N A254469 Sixth partial sums of cubes (A000578).
%H A254469 Luciano Ancora, <a href="/A254469/b254469.txt">Table of n, a(n) for n = 1..1000</a>
%H A254469 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>.
%H A254469 Luciano Ancora, <a href="/A254647/a254647_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>.
%H A254469 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 15.
%H A254469 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A254469 G.f.: (x + 4*x^2 + x^3)/(- 1 + x)^10.
%F A254469 a(n) = n*(1 + n)^2*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(6 + n)/60480.
%F A254469 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^3.
%F A254469 From _Amiram Eldar_, Jan 26 2022: (Start)
%F A254469 Sum_{n>=1} 1/a(n) = 217/200.
%F A254469 Sum_{n>=1} (-1)^(n+1)/a(n) = 223769/200 - 8064*log(2)/5. (End)
%e A254469 First differences: 1, 7, 19, 37, 61, 91, ... (A003215)
%e A254469 -------------------------------------------------------------------------
%e A254469 The cubes: 1, 8, 27, 64, 125, 216, ... (A000578)
%e A254469 -------------------------------------------------------------------------
%e A254469 First partial sums: 1, 9, 36, 100, 225, 441, ... (A000537)
%e A254469 Second partial sums: 1, 10, 46, 146, 371, 812, ... (A024166)
%e A254469 Third partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094)
%e A254469 Fourth partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097)
%e A254469 Fifth partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102)
%e A254469 Sixth partial sums: 1, 14, 96, 450, 1650, 5082, ... (this sequence)
%t A254469 Table[n (1 + n)^2 (2 + n) (3 + n) (4 + n) (5 + n)^2 (6 + n)/60480, {n, 27}] (* or *) CoefficientList[Series[(1 + 4 x + x^2)/(- 1 + x)^10, {x, 0, 26}], x]
%t A254469 Nest[Accumulate,Range[30]^3,6] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,14,96,450,1650,5082,13728,33462,75075,157300},30] (* _Harvey P. Dale_, Sep 03 2016 *)
%o A254469 (Magma) [n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480: n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2015
%o A254469 (PARI) a(n)=n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A254469 Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254470, A254471, A254472.
%K A254469 nonn,easy
%O A254469 1,2
%A A254469 _Luciano Ancora_, Feb 15 2015