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n(n+1)(n^2 -3n + 6)/8.
(Formerly M2794)
5

%I M2794 #34 May 27 2023 18:58:33

%S 1,3,9,25,60,126,238,414,675,1045,1551,2223,3094,4200,5580,7276,9333,

%T 11799,14725,18165,22176,26818,32154,38250,45175,53001,61803,71659,

%U 82650,94860,108376,123288,139689,157675,177345,198801,222148,247494,274950

%N n(n+1)(n^2 -3n + 6)/8.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A004255/b004255.txt">Table of n, a(n) for n = 1..1000</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: -x*(1-2*x+4*x^2) / (x-1)^5. - _Simon Plouffe_ in his 1992 dissertation.

%t lst={};Do[AppendTo[lst, n*(n+1)*(n^2-3*n+6)/8], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 19 2008 *)

%t Table[n(n+1)(n^2-3n+6)/8,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,9,25,60},40] (* _Harvey P. Dale_, May 27 2023 *)

%o (Magma) [n*(n+1)*(n^2-3*n+6)/8: n in [1..50]]; // _Vincenzo Librandi_, Jun 07 2013

%o (PARI) a(n)=n*(n+1)*(n^2-3*n+6)/8 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Partial sums of A060354. Equals (1/2) A062026.

%K nonn,easy

%O 1,2

%A Dennis S. Kluk (mathemagician(AT)ameritech.net)