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%I #38 Sep 08 2022 08:44:48
%S 4,5,16,19,40,46,80,90,140,155,224,245,336,364,480,516,660,705,880,
%T 935,1144,1210,1456,1534,1820,1911,2240,2345,2720,2840,3264,3400,3876,
%U 4029,4560,4731,5320,5510,6160,6370,7084,7315,8096,8349,9200,9476,10400,10700,11700
%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = (natural numbers), t = (natural numbers >= 3).
%H G. C. Greubel, <a href="/A024854/b024854.txt">Table of n, a(n) for n = 2..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = Sum_{i=1..floor(n/2)} i*(n-i+3) = -floor(n/2)*(floor(n/2)+1)*(2*floor(n/2)-3n-8)/6. - _Wesley Ivan Hurt_, Sep 20 2013
%F G.f. x^2*(4 + x - x^2) / ( (1+x)^3*(1-x)^4 ). - _R. J. Mathar_, Sep 25 2013
%F a(n) = 4*A058187(n-2) + A058187(n-3) - A058187(n-4). - _R. J. Mathar_, Sep 25 2013
%F a(n) = (4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48. - _Luce ETIENNE_, Nov 14 2014
%F E.g.f.: (1/24)*( x*(9 + 18*x + 2*x^2)*cosh(x) + (-9 + 30*x + 15*x^2 + 2*x^3)*sinh(x) ). - _G. C. Greubel_, Jul 13 2022
%p seq(sum(i*(k-i+3), i=1..floor(k/2)), k=2..70); # _Wesley Ivan Hurt_, Sep 20 2013
%t Table[-Floor[n/2] * (Floor[n/2] + 1) * (2 * Floor[n/2] - 3n - 8)/6, {n, 2, 100}] (* _Wesley Ivan Hurt_, Sep 20 2013 *)
%t CoefficientList[Series[- (- 4 - x + x^2)/((1 + x)^3 (x - 1)^4), {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 31 2014 *)
%o (Magma) [(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48: n in [2..60]]; // _Vincenzo Librandi_, Oct 31 2014
%o (SageMath) [(4*n^3+21*n^2+14*n-9+3*(n^2+6*n+3)*(-1)^n)/48 for n in (2..60)] # _G. C. Greubel_, Jul 13 2022
%Y Cf. A023855, A023856, A023857, A024305, A058187.
%K nonn,easy
%O 2,1
%A _Clark Kimberling_
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