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%I #31 Sep 08 2022 08:45:50
%S 0,1,41,212,660,1585,3241,5936,10032,15945,24145,35156,49556,67977,
%T 91105,119680,154496,196401,246297,305140,373940,453761,545721,650992,
%U 770800,906425,1059201,1230516,1421812,1634585,1870385,2130816,2417536
%N a(n) = n*(27*n^3 + 22*n^2 - 21*n - 16)/12.
%C The sequence is related to A172082 by a(n) = n*A172082(n)-sum(A172082(i), i=0..n-1).
%C This is the case d=9 in the identity n^2*(n+1)*(2*d*n -2*d +3)/6 - Sum_{k=0..n-1} k*(k+1)*(2*d*k -2*d +3)/6 = n*(n+1)*(3*d*n^2 - d*n + 4*n - 2*d + 2)/12. - _Bruno Berselli_, May 07 2010, Jan 28 2011
%H Vincenzo Librandi, <a href="/A172085/b172085.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n*(n+1)*(27*n^2 -5*n -16)/12.
%F From _Bruno Berselli_, Jan 28 2011: (Start)
%F G.f.: x*(1 +36*x +17*x^2)/(1-x)^5.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F E.g.f.: x*(12 + 234*x + 184*x^2 + 27*x^3)*exp(x)/12. - _G. C. Greubel_, Aug 30 2019
%p seq(n*(n+1)*(27*n^2 -5*n -16)/12, n=0..40); # _G. C. Greubel_, Aug 30 2019
%t CoefficientList[Series[x(1 +36x +17x^2)/(1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 02 2014 *)
%t Table[n*(n+1)*(27*n^2-5*n-16)/12, {n,0,40}] (* _G. C. Greubel_, Aug 30 2019 *)
%o (Magma) [n*(n+1)*(27*n^2-5*n-16)/12: n in [0..40]]; // _Vincenzo Librandi_, Jan 02 2014
%o (PARI) vector(40, n, m=n-1; n*m*(27*m^2 -5*m -16)/12) \\ _G. C. Greubel_, Aug 30 2019
%o (Sage) [n*(n+1)*(27*n^2 -5*n -16)/12: n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o (GAP) List([0..40], n-> n*(n+1)*(27*n^2 -5*n -16)/12); # _G. C. Greubel_, Aug 30 2019
%Y Cf. A172082.
%K nonn,easy
%O 0,3
%A _Vincenzo Librandi_, Jan 25 2010
%E Librandi's contribution restored and rewritten from _Bruno Berselli_, Feb 29 2012
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