A206809 a(n) = Sum_{0

7, 52, 208, 608, 1463, 3080, 5880, 10416, 17391, 27676, 42328, 62608, 89999, 126224, 173264, 233376, 309111, 403332, 519232, 660352, 830599, 1034264, 1276040, 1561040, 1894815, 2283372, 2733192, 3251248, 3845023, 4522528, 5292320

Offset: 2

Author

Clark Kimberling, Feb 15 2012

Keywords

• nonn
• easy

Comments

Partial sums of A206808. For a guide to related sequences, see A206817.

Examples

			a(3) = (8-1) + (27-1) + (27-8) = 52.
a(4) = a(3) + (64-1) + (64-8) + (64-27) = 208.
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 2..10000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A206808, A206817.

Programs

Mathematica

s[k_] := k^3; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]  (* A206808 *)
Flatten[Table[t[n], {n, 2, 35}]]  (* A206809 *)

PARI

vector(100, n, n*(9*n^4+60*n^3+145*n^2+150*n+56)/60) \\ Colin Barker, Jul 11 2014

PARI

Vec(x^2*(x^2+10*x+7)/(x-1)^6 + O(x^100)) \\ Colin Barker, Jul 11 2014

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

Formula

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

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