A099903 Sum of all matrix elements of n X n matrix M(i,j) = i^3+j^3, (i,j = 1..n). a(n) = n^3*(n+1)^2/2.

2, 36, 216, 800, 2250, 5292, 10976, 20736, 36450, 60500, 95832, 146016, 215306, 308700, 432000, 591872, 795906, 1052676, 1371800, 1764000, 2241162, 2816396, 3504096, 4320000, 5281250, 6406452, 7715736, 9230816, 10975050, 12973500

Offset

1,1

Comments

Numerator of a(n)/n! is A099904(n).

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = Sum_{i=1..n, j=1..n} (i^3 + j^3).

a(n) = 2*n*Sum_{k=1..n} k^3. - Gary Detlefs, Oct 26 2011

a(n) = (n^5 + 2*n^4 + n^3)/2. - Charles R Greathouse IV, Oct 27 2011

G.f.: 2*x*(1+12*x+15*x^2+2*x^3)/(1-x)^6. - Colin Barker, May 04 2012

From Amiram Eldar, Nov 02 2021: (Start)

Sum_{n>=1} 1/a(n) = 2*zeta(3) - Pi^2 + 8.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/2 + 12*log(2) - Pi^2/6 - 8. (End)

Example

a(3) = (1/2) * (2^3)*(2+1)^2 = 36.
  (or)
a(3) = (1^3+1^3) + (1^3+2^3) + (2^3+1^3) + (2^3+2^3) = 36.

Maple

A099903:=n->(n^5+2*n^4+n^3)/2; seq(A099903(n), n=1..30); # Wesley Ivan Hurt, Feb 26 2014

Mathematica

Table[(n^5+2*n^4+n^3)/2, {n, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)

Prog

(PARI) a(n)=(n^5+2*n^4+n^3)/2 \\ Charles R Greathouse IV, Oct 27 2011
(Magma) [(n^5+2*n^4+n^3)/2: n in [1..30]]; // Wesley Ivan Hurt, May 25 2014

Crossrefs

Cf. A099904, A019584, A098077.

Sequence in context: A206688 A226419 A025531 * A341535 A074426 A082636

Adjacent sequences: A099900 A099901 A099902 * A099904 A099905 A099906

Keyword

nonn,easy

Author

Alexander Adamchuk, Oct 29 2004

Status

approved