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A129936 a(n) = (n-2)*(n+3)*(n+2)/6.

%I A129936 #38 Aug 04 2025 21:16:31
%S A129936 -2,-2,0,5,14,28,48,75,110,154,208,273,350,440,544,663,798,950,1120,
%T A129936 1309,1518,1748,2000,2275,2574,2898,3248,3625,4030,4464,4928,5423,
%U A129936 5950,6510,7104,7733,8398,9100,9840,10619,11438,12298,13200,14145,15134,16168,17248
%N A129936 a(n) = (n-2)*(n+3)*(n+2)/6.
%C A129936 Essentially the same as A005586.
%H A129936 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A129936 a(n) = binomial(n+3,3) - (n + 3)*(n + 2)/2.
%F A129936 a(n) = A214292(n+2,2). - _Reinhard Zumkeller_, Jul 12 2012
%F A129936 G.f.: (x^3-4*x^2+6*x-2)/(x-1)^4. - _Colin Barker_, Sep 05 2012
%F A129936 From _Wesley Ivan Hurt_, Sep 21 2013: (Start)
%F A129936 a(n) = Sum_{i=1..n+2} i*(n-i+1).
%F A129936 a(n+2) = A000292(n+1) + A034856(n), n>0. (End)
%F A129936 From _Amiram Eldar_, Sep 26 2022: (Start)
%F A129936 Sum_{n>=3} 1/a(n) = 77/200.
%F A129936 Sum_{n>=3} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
%F A129936 From _Elmo R. Oliveira_, Aug 04 2025: (Start)
%F A129936 E.g.f.: exp(x)*(x^3 + 6*x^2 - 12)/6.
%F A129936 a(n) = A023444(n)*A002378(n+2)/6.
%F A129936 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%p A129936 seq(sum(i*(k-i+1), i=1..k+2), k=0..99); # _Wesley Ivan Hurt_, Sep 21 2013
%t A129936 f[n_] = Binomial[n + 3, 3] - (n + 3)*(n + 2)/2; Table[f[n], {n, 0, 30}]
%t A129936 LinearRecurrence[{4,-6,4,-1},{-2,-2,0,5},50] (* _Harvey P. Dale_, Jul 03 2020 *)
%o A129936 (PARI) a(n)=(n-2)*(n+3)*(n+2)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A129936 Cf. A000292, A002378, A005586, A023444, A034856, A214292.
%K A129936 easy,sign
%O A129936 0,1
%A A129936 _Roger L. Bagula_, Jun 09 2007
%E A129936 More terms from _Wesley Ivan Hurt_, Sep 21 2013