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 A254644 #51 Sep 08 2022 08:46:11
%S A254644 1,36,381,2336,10326,36552,110022,292512,704847,1567852,3263403,
%T A254644 6422208,12046268,21675408,37608828,63194304,103199469,164281524,
%U A254644 255573769,389409504,582206130,855534680,1237402530,1763779680,2480401755,3444885756,4729197591,6422513536,8634521016,11499207456
%N A254644 Fourth partial sums of fifth powers (A000584).
%H A254644 Luciano Ancora, <a href="/A254644/b254644.txt">Table of n, a(n) for n = 1..1000</a>
%H A254644 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>
%H A254644 Luciano Ancora, <a href="/A254644/a254644_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>
%H A254644 <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 A254644 G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^10.
%F A254644 a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(-24 + 20*n + 85*n^2 + 40*n^3 + 5*n^4)/15120.
%F A254644 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^5.
%e A254644 Fifth differences: 1, 27, 93, 119, 120, (repeat 120) (A101100)
%e A254644 Fourth differences: 1, 28, 121, 240, 360, 480, ... (A101095)
%e A254644 Third differences: 1, 29, 150, 390, 750, 1230, ... (A101096)
%e A254644 Second differences: 1, 30, 180, 570, 1320, 2550, ... (A101098)
%e A254644 First differences: 1, 31, 211, 781, 2101, 4651, ... (A022521)
%e A254644 -------------------------------------------------------------------------
%e A254644 The fifth powers: 1, 32, 243, 1024, 3125, 7776, ... (A000584)
%e A254644 -------------------------------------------------------------------------
%e A254644 First partial sums: 1, 33, 276, 1300, 4425, 12201, ... (A000539)
%e A254644 Second partial sums: 1, 34, 310, 1610, 6035, 18236, ... (A101092)
%e A254644 Third partial sums: 1, 35, 345, 1955, 7990, 26226, ... (A101099)
%e A254644 Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (this sequence)
%p A254644 seq(binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t A254644 Table[n(1+n)(2+n)(3+n)(4+n)(-24 +20n +85n^2 +40n^3 +5n^4)/15120, {n, 30}] (* or *) Accumulate[Accumulate[Accumulate[Accumulate[Range[24]^5]]]] (* or *) CoefficientList[Series[(1 +26x +66x^2 +26x^3 +x^4)/(1-x)^10, {x, 0, 30}], x]
%t A254644 Nest[Accumulate,Range[30]^5,4] (* or *) LinearRecurrence[{10,-45,120, -210,252,-210,120,-45,10,-1}, {1,36,381,2336,10326,36552,110022,292512, 704847,1567852},30] (* _Harvey P. Dale_, May 08 2016 *)
%o A254644 (PARI) vector(30, n, m=n+2; binomial(m+2,5)*(5*m^4 -35*m^2 +36)/126) \\ _G. C. Greubel_, Aug 28 2019
%o A254644 (Magma) [Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019
%o A254644 (Sage) [binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019
%o A254644 (GAP) List([1..30], n-> Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126); # _G. C. Greubel_, Aug 28 2019
%Y A254644 Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100.
%Y A254644 Cf. A101091 (fourth partial sums of fourth powers).
%K A254644 nonn,easy
%O A254644 1,2
%A A254644 _Luciano Ancora_, Feb 05 2015
%E A254644 Edited by _Bruno Berselli_, Feb 10 2015