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 A253636 #63 Jan 28 2025 10:59:37
%S A253636 1,258,7076,79430,542409,2685004,10592400,35277012,103008345,
%T A253636 270739678,652829892,1464901802,3092704433,6196296120,11862778432,
%U A253636 21824228040,38761435089,66718602714,111659333380,182200064046,290563654073,453803117636,695353566480,1046979329500
%N A253636 Second partial sums of eighth powers (A001016).
%C A253636 The general formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m)-F(m+1), where F(m) are the m-th Faulhaber’s formulas.
%H A253636 G. C. Greubel, <a href="/A253636/b253636.txt">Table of n, a(n) for n = 1..1000</a>
%H A253636 Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>
%H A253636 Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>
%H A253636 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 A253636 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A253636 a(n) = (2*n^10 + 20*n^9 + 75*n^8 + 120*n^7 + 42*n^6 - 84*n^5 - 50*n^4 + 40*n^3 + 21*n^2 - 6*n)/180.
%F A253636 a(n) = 2*a(n-1) - a(n-2) + n^8. - _Robert Israel_, Jan 07 2015
%F A253636 G.f.: x*(1 + x)*(1 + 246*x + 4047*x^2 + 11572*x^3 + 4047*x^4 + 246*x^5 + x^6) / (1 - x)^11. - _Bruno Berselli_, Jan 08 2015
%p A253636 seq(n*(n+1)^2*(n+2)*(2*n^6 +12*n^5 +17*n^4 -12*n^3 -19*n^2 +18*n -3))/180, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t A253636 Table[n(n+1)^2(n+2)(2n^6 +12n^5 +17n^4 -12n^3 -19n^2 +18n -3)/180, {n,30}] (* _Bruno Berselli_, Jan 08 2015 *)
%t A253636 Nest[Accumulate,Range[30]^8,2] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,258,7076,79430,542409,2685004,10592400, 35277012, 103008345,270739678,652829892},30] (* _Harvey P. Dale_, Jul 02 2017 *)
%o A253636 (Sage)
%o A253636 [(2*n^10+20*n^9+75*n^8+120*n^7+42*n^6-84*n^5-50*n^4+40*n^3+21*n^2-6*n)/180 for n in [1..24]] # _Tom Edgar_, Jan 07 2015
%o A253636 (Magma) [n*(n+1)^2*(n+2)*(2*n^6+12*n^5+17*n^4-12*n^3-19*n^2+18*n-3)/180: n in [1..25]]; // _Bruno Berselli_, Jan 08 2015
%o A253636 (PARI) a(n)=(2*n^10+20*n^9+75*n^8+120*n^7+42*n^6-84*n^5-50*n^4+40*n^3+21*n^2-6*n)/180 \\ _Charles R Greathouse IV_, Sep 08 2015
%o A253636 (GAP) List([1..30], n-> n*(n+1)^2*(n+2)*(2*n^6 +12*n^5 +17*n^4 -12*n^3 -19*n^2 +18*n -3)/180); # _G. C. Greubel_, Aug 28 2019
%Y A253636 Cf. A001016, A101089, A101092, A101093.
%K A253636 nonn,easy
%O A253636 1,2
%A A253636 _Luciano Ancora_, Jan 07 2015