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 A101092 #68 Jan 14 2025 11:06:51
%S A101092 1,34,310,1610,6035,18236,47244,109020,229845,450670,832546,1463254,
%T A101092 2465255,4005080,6304280,9652056,14419689,21076890,30210190,42543490,
%U A101092 58960891,80531924,108539300,144509300,190244925,247861926,319827834,409004110,518691535
%N A101092 Second partial sums of fifth powers (A000584).
%H A101092 Vincenzo Librandi, <a href="/A101092/b101092.txt">Table of n, a(n) for n = 1..1000</a>
%H A101092 Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>
%H A101092 Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>
%H A101092 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. 13.
%H A101092 C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014 (But beware of a typo)
%H A101092 Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]
%H A101092 Cecilia Rossiter, <a href="/A101092/a101092_1.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]
%H A101092 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A101092 a(n) = (n*(1 + n)*(2 + n)*(-1 + n*(2 + n))*(1 + 2*n*(2 + n)))/84.
%F A101092 G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^8. - _Colin Barker_, Apr 16 2012
%F A101092 a(n) = Sum_{i=1..n} i*(n+1-i)^5, by the definition. - _Bruno Berselli_, Jan 31 2014
%F A101092 a(n) = 2*a(n-1) - a(n-2) + n^5. - _Luciano Ancora_, Jan 08 2015
%F A101092 E.g.f.: exp(x)*x*(84 + 1344*x + 2954*x^2 + 1995*x^3 + 525*x^4 + 56*x^5 + 2*x^6)/84. - _Stefano Spezia_, May 04 2024
%p A101092 f:=n->(2*n^7-7*n^5+7*n^3-2*n)/84;
%p A101092 [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014
%t A101092 CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%t A101092 Nest[Accumulate,Range[30]^5,2] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,34,310,1610,6035,18236,47244,109020},30] (* _Harvey P. Dale_, Aug 22 2022 *)
%o A101092 (Magma) [(n*(1+n)*(2+n)*(-1+n*(2+n))*(1+2*n*(2+n)))/84: n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014
%o A101092 (Sage) [n*(1+n)*(2+n)*(n*(2+n)-1)*(1+2*n*(2+n))/84 for n in range(1,30)] # _Danny Rorabaugh_, Apr 21 2015
%o A101092 (PARI) a(n)=n*(n+1)*(n+2)*(n*(n+2)-1)*(2*n*(2 + n)+1)/84 \\ _Charles R Greathouse IV_, Apr 21 2015
%o A101092 (Python)
%o A101092 def A101092(n): return n*(n*(n*(n*(n*(n*(n+7<<1)+35)+35)+7)-7)-2)//84 # _Chai Wah Wu_, Oct 03 2024
%Y A101092 Cf. A000539, A101099.
%Y A101092 Cf. A254640.
%K A101092 nonn,easy
%O A101092 1,2
%A A101092 Cecilia Rossiter, Dec 14 2004
%E A101092 Edited by _Ralf Stephan_, Dec 16 2004