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%I #61 Sep 08 2022 08:45:15
%S 1,66,860,5750,26265,93436,278256,725220,1703625,3682030,7431996,
%T 14167946,25730705,44823000,75305920,122566056,193963761,299373690,
%U 451829500,668285310,970507241,1386109076,1949746800,2704487500
%N Second partial sums of sixth powers (A001014).
%H Vincenzo Librandi, <a href="/A101093/b101093.txt">Table of n, a(n) for n = 1..1000</a>
%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>
%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>
%H 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
%H C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = n*(1 + n)^2*(2 + n)*(-1 + n*(2 + n))*(-2 + 3*n*(2 + n))/168.
%F G.f.: x*(1+x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4)/(1-x)^9. - _Colin Barker_, Dec 18 2012
%F a(n) = Sum_{i=1..n} i*(n+1-i)^6, by the definition. - _Bruno Berselli_, Jan 31 2014
%F a(n) = 2*a(n-1) - a(n-2) + n^6. - _Luciano Ancora_, Jan 08 2015
%p f:=n->(3*n^8-14*n^6+21*n^4-10*n^2)/168;
%p [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014
%t CoefficientList[Series[(x+1)(x^4+56x^3+246x^2+56x+1)/(1-x)^9, {x,0,40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%t Nest[Accumulate,Range[30]^6,2] (* or *) LinearRecurrence[{9,-36,84,-126, 126,-84,36,-9,1},{1,66,860,5750,26265,93436,278256,725220,1703625},30] (* _Harvey P. Dale_, Jun 05 2019 *)
%o (Magma) [n*(1+n)^2*(2+n)*(-1+n*(2+n))*(-2+3*n*(2+n))/168: n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014
%o (PARI) vector(30, n, m=n+1; m^2*(3*m^6 -14*m^4 +21*m^2 -10)/168) \\ _G. C. Greubel_, Aug 28 2019
%o (Sage) [n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168 for n in (2..30)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) List([2..30], n-> n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A000540.
%K nonn,easy
%O 1,2
%A Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%E Edited by _Ralf Stephan_, Dec 16 2004
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