A206811 - OEIS

%I #19 Jun 17 2017 03:55:10

%S 15,160,830,2976,8477,20608,44604,88320,162987,284064,472186,754208,

%T 1164345,1745408,2550136,3642624,5099847,7013280,9490614,12657568,

%U 16659797,21664896,27864500,35476480,44747235,55954080,69407730

%N Sum_{0<j<k<=n} (k^4-j^4).

%C Partial sums of A206810.  For a guide to related sequences, see A206817.

%H Danny Rorabaugh, <a href="/A206811/b206811.txt">Table of n, a(n) for n = 2..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (n*(1+n)^2*(1-6*n+n^2+4*n^3))/30. G.f.: -x^2*(x^3+25*x^2+55*x+15) / (x-1)^7. - _Colin Barker_, Jul 11 2014

%e a(4) = 16-1 + 81-1 + 81-16 = 160.

%t s[k_] := k^4; t[1] = 0;

%t p[n_] := Sum[s[k], {k, 1, n}];

%t c[n_] := n*s[n] - p[n];

%t t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]

%t Table[c[n], {n, 2, 50}]  (* A206810  *)

%t Flatten[Table[t[n], {n, 2, 35}]] (* A206811 *)

%o (PARI) Vec(-x^2*(x^3+25*x^2+55*x+15)/(x-1)^7 + O(x^100)) \\ _Colin Barker_, Jul 11 2014

%o (Sage) [sum([sum([k^4-j^4 for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,29)] # _Danny Rorabaugh_, Apr 18 2015

%Y Cf. A206810, A206817.

%K nonn,easy

%O 2,1

%A _Clark Kimberling_, Feb 15 2012