|
%I #30 Sep 08 2022 08:44:49
%S 13,28,50,80,119,168,228,300,385,484,598,728,875,1040,1224,1428,1653,
%T 1900,2170,2464,2783,3128,3500,3900,4329,4788,5278,5800,6355,6944,
%U 7568,8228,8925,9660,10434,11248,12103,13000,13940,14924,15953,17028,18150,19320,20539,21808,23128,24500,25925
%N dot product (n,n-1,...2,1).(3,4,...,n,1,2).
%H Vincenzo Librandi, <a href="/A026054/b026054.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A023551(n+1) + 4.
%F From _Colin Barker_, Sep 17 2012: (Start)
%F a(n) = n*(n^2+9*n-10)/6.
%F G.f.: x^3*(13 - 24*x + 16*x^2 - 4*x^3)/(1-x)^4. (End)
%F E.g.f.: x^2*(-12 + (12+x)*exp(x))/6. - _G. C. Greubel_, Oct 30 2019
%p seq(n*(n^2+9*n-10)/6, n=3..60); # _G. C. Greubel_, Oct 30 2019
%t Table[Range[n,1,-1].RotateLeft[Range[n],2],{n,3,60}] (* or *) LinearRecurrence[ {4,-6,4,-1},{13,28,50,80},60] (* _Harvey P. Dale_, Oct 14 2012 *)
%t Drop[CoefficientList[Series[x(13 -24x +16x^2 -4x^3)/(1-x)^4, {x, 0, 60}], x], 1] (* _Vincenzo Librandi_, Oct 17 2013 *)
%o (Magma) [n*(n^2+9*n-10)/6: n in [3..60]]; // _Vincenzo Librandi_, Oct 17 2013
%o (PARI) vector(60, n, (n+2)*((n+2)^2+9*(n+2)-10)/6) \\ _G. C. Greubel_, Oct 30 2019
%o (Magma) [n*(n^2+9*n-10)/6: n in [0..60]]; // _G. C. Greubel_, Oct 30 2019
%o (Sage) [n*(n^2+9*n-10)/6 for n in (0..60)] # _G. C. Greubel_, Oct 30 2019
%o (GAP) List([0..60], n-> n*(n^2+9*n-10)/6); # _G. C. Greubel_, Oct 30 2019
%Y Cf. A023551.
%Y Column 2 of triangle A094415.
%Y Essentially the same as A060488. - _Vladeta Jovovic_, Jun 15 2006
%K nonn,easy
%O 3,1
%A _Clark Kimberling_
%E Closed-form formula corrected by _Colin Barker_, Sep 17 2012
|
