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A117662 a(n) = n*(n-1)*(n-2)*(n+3)/12.

%I A117662 #43 May 17 2025 05:41:41
%S A117662 0,0,0,3,14,40,90,175,308,504,780,1155,1650,2288,3094,4095,5320,6800,
%T A117662 8568,10659,13110,15960,19250,23023,27324,32200,37700,43875,50778,
%U A117662 58464,66990,76415,86800,98208,110704,124355,139230,155400,172938,191919
%N A117662 a(n) = n*(n-1)*(n-2)*(n+3)/12.
%C A117662 Also, the number of external intersections of the diagonals of a general n-gon = (A176145) - (A000332). - _Michel Lagneau_, Apr 21 2010
%H A117662 Vincenzo Librandi, <a href="/A117662/b117662.txt">Table of n, a(n) for n = 0..1000</a>
%H A117662 Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan, and Chelsea Sato, <a href="https://arxiv.org/abs/2505.06486">The Chromatic Symmetric Function for Unicyclic Graphs</a>, arXiv:2505.06486 [math.CO], 2025. See p. 12.
%H A117662 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A117662 G.f.: x^3*(3-x)/(1-x)^5. - _Colin Barker_, Jan 31 2012
%F A117662 From _Amiram Eldar_, May 17 2025: (Start)
%F A117662 Sum_{n>=3} 1/a(n) = 137/300.
%F A117662 Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/5 - 1247/300. (End)
%p A117662 seq(n*(n-1)*(n-2)*(n+3)/12, n=0..40); # _Wesley Ivan Hurt_, Oct 10 2013
%t A117662 Table[n(n-1)(n-2)(n+3)/12, {n,0,100}] (* _Wesley Ivan Hurt_, Sep 26 2013 *)
%t A117662 CoefficientList[Series[x^3 (3 - x)/(1 - x)^5, {x, 0, 80}], x] (* _Vincenzo Librandi_, Oct 10 2013 *)
%t A117662 LinearRecurrence[{5,-10,10,-5,1},{0,0,0,3,14},80] (* _Harvey P. Dale_, Jan 01 2025 *)
%o A117662 (Magma) [n*(n-1)*(n-2)*(n+3)/12: n in [0..50]]; // _Vincenzo Librandi_, Oct 10 2013
%Y A117662 Essentially the same as A050297 and A005701.
%Y A117662 Cf. A000332, A176145.
%K A117662 nonn,easy
%O A117662 0,4
%A A117662 _Roger L. Bagula_, Apr 11 2006
%E A117662 Edited by _N. J. A. Sloane_, Apr 23 2006