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 A254681 #46 Jan 26 2022 02:29:29
%S A254681 1,21,176,936,3750,12342,35112,89232,207207,446875,906048,1743248,
%T A254681 3206268,5670588,9690000,16062144,25912029,40797009,62837104,94875000,
%U A254681 140670530,205134930,294610680,417203280,583171875,805386231
%N A254681 Fifth partial sums of fourth powers (A000583).
%H A254681 Luciano Ancora, <a href="/A254681/b254681.txt">Table of n, a(n) for n = 1..1000</a>
%H A254681 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>.
%H A254681 Luciano Ancora, <a href="/A254647/a254647_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>.
%H A254681 <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 A254681 G.f.:(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^10.
%F A254681 a(n) = n^2*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(5 + 2*n)/30240.
%F A254681 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^4.
%F A254681 E.g.f.: (1/30240)*exp(x)*(30240 + 604800*x + 2041200*x^2 + 2368800*x^3 + 1233540*x^4 + 326592*x^5 + 46410*x^6 + 3540*x^7 + 135*x^8 + 2*x^9). - _Stefano Spezia_, Dec 02 2018
%F A254681 From _Amiram Eldar_, Jan 26 2022: (Start)
%F A254681 Sum_{n>=1} 1/a(n) = 172032*log(2)/125 - 2382233/2500.
%F A254681 Sum_{n>=1} (-1)^(n+1)/a(n) = 42*Pi^2/25 - 43008*Pi/125 + 2663213/2500. (End)
%e A254681 Fourth differences: 1, 12, 23, 24, (repeat 24) ... (A101104)
%e A254681 Third differences: 1, 13, 36, 60, 84, 108, ... (A101103)
%e A254681 Second differences: 1, 14, 50, 110, 194, 302, ... (A005914)
%e A254681 First differences: 1, 15, 65, 175, 369, 671, ... (A005917)
%e A254681 -------------------------------------------------------------------------
%e A254681 The fourth powers: 1, 16, 81, 256, 625, 1296, ... (A000583)
%e A254681 -------------------------------------------------------------------------
%e A254681 First partial sums: 1, 17, 98, 354, 979, 2275, ... (A000538)
%e A254681 Second partial sums: 1, 18, 116, 470, 1449, 3724, ... (A101089)
%e A254681 Third partial sums: 1, 19, 135, 605, 2054, 5778, ... (A101090)
%e A254681 Fourth partial sums: 1, 20, 155, 760, 2814, 8592, ... (A101091)
%e A254681 Fifth partial sums: 1, 21, 176, 936, 3750, 12342, ... (this sequence)
%p A254681 seq(coeff(series((x+11*x^2+11*x^3+x^4)/(1-x)^10,x,n+1), x, n), n = 1 .. 30); # _Muniru A Asiru_, Dec 02 2018
%t A254681 Table[n^2(1+n)(2+n)(3+n)(4+n)(5+n)^2(5+2n)/30240, {n,26}] (* or *)
%t A254681 CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(1-x)^10, {x,0,25}], x]
%t A254681 CoefficientList[Series[(1/30240)E^x (30240 + 604800 x + 2041200 x^2 + 2368800 x^3 + 1233540 x^4 + 326592 x^5 + 46410 x^6 + 3540 x^7 + 135 x^8 + 2 x^9), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* _Stefano Spezia_, Dec 02 2018 *)
%t A254681 Nest[Accumulate[#]&,Range[30]^4,5] (* _Harvey P. Dale_, Jan 03 2022 *)
%o A254681 (PARI) my(x='x+O('x^30)); Vec((x+11*x^2+11*x^3+x^4)/(1-x)^10) \\ _G. C. Greubel_, Dec 01 2018
%o A254681 (Magma) [Binomial(n+5,6)*n*(n+5)*(2*n+5)/42: n in [1..30]]; // _G. C. Greubel_, Dec 01 2018
%o A254681 (Sage) [binomial(n+5,6)*n*(n+5)*(2*n+5)/42 for n in (1..30)] # _G. C. Greubel_, Dec 01 2018
%Y A254681 Cf. A000538, A000583, A005914, A005917, A101089, A101090, A101091, A101103, A101104, A254682, A254683, A254684.
%K A254681 nonn,easy
%O A254681 1,2
%A A254681 _Luciano Ancora_, Feb 12 2015