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A206809 a(n) = Sum_{0

%I A206809 #17 Jul 23 2025 22:45:35
%S A206809 7,52,208,608,1463,3080,5880,10416,17391,27676,42328,62608,89999,
%T A206809 126224,173264,233376,309111,403332,519232,660352,830599,1034264,
%U A206809 1276040,1561040,1894815,2283372,2733192,3251248,3845023,4522528,5292320
%N A206809 a(n) = Sum_{0<j<k<=n} k^3-j^3.
%C A206809 Partial sums of A206808.  For a guide to related sequences, see A206817.
%H A206809 Danny Rorabaugh, <a href="/A206809/b206809.txt">Table of n, a(n) for n = 2..10000</a>
%H A206809 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A206809 a(n) = (n*(-4-15*n-5*n^2+15*n^3+9*n^4))/60. G.f.: x^2*(x^2+10*x+7) / (x-1)^6. - _Colin Barker_, Jul 11 2014
%F A206809 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Wesley Ivan Hurt_, Jul 23 2025
%e A206809 a(3) = (8-1) + (27-1) + (27-8) = 52.
%e A206809 a(4) = a(3) + (64-1) + (64-8) + (64-27) = 208.
%t A206809 s[k_] := k^3; t[1] = 0;
%t A206809 p[n_] := Sum[s[k], {k, 1, n}];
%t A206809 c[n_] := n*s[n] - p[n];
%t A206809 t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
%t A206809 Table[c[n], {n, 2, 50}]  (* A206808 *)
%t A206809 Flatten[Table[t[n], {n, 2, 35}]]  (* A206809 *)
%o A206809 (PARI) vector(100, n, n*(9*n^4+60*n^3+145*n^2+150*n+56)/60) \\ _Colin Barker_, Jul 11 2014
%o A206809 (PARI) Vec(x^2*(x^2+10*x+7)/(x-1)^6 + O(x^100)) \\ _Colin Barker_, Jul 11 2014
%o A206809 (Sage) [sum([sum([k^3-j^3 for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,33)] # _Danny Rorabaugh_, Apr 18 2015
%Y A206809 Cf. A206808, A206817.
%K A206809 nonn,easy
%O A206809 2,1
%A A206809 _Clark Kimberling_, Feb 15 2012