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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.

%I A099903 #36 Sep 08 2022 08:45:15
%S A099903 2,36,216,800,2250,5292,10976,20736,36450,60500,95832,146016,215306,
%T A099903 308700,432000,591872,795906,1052676,1371800,1764000,2241162,2816396,
%U A099903 3504096,4320000,5281250,6406452,7715736,9230816,10975050,12973500
%N 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.
%C A099903 Numerator of a(n)/n! is A099904(n).
%H A099903 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A099903 a(n) = Sum_{i=1..n, j=1..n} (i^3 + j^3).
%F A099903 a(n) = 2*n*Sum_{k=1..n} k^3. - _Gary Detlefs_, Oct 26 2011
%F A099903 a(n) = (n^5 + 2*n^4 + n^3)/2. - _Charles R Greathouse IV_, Oct 27 2011
%F A099903 G.f.: 2*x*(1+12*x+15*x^2+2*x^3)/(1-x)^6. - _Colin Barker_, May 04 2012
%F A099903 From _Amiram Eldar_, Nov 02 2021: (Start)
%F A099903 Sum_{n>=1} 1/a(n) = 2*zeta(3) - Pi^2 + 8.
%F A099903 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/2 + 12*log(2) - Pi^2/6 - 8. (End)
%e A099903 a(3) = (1/2) * (2^3)*(2+1)^2 = 36.
%e A099903   (or)
%e A099903 a(3) = (1^3+1^3) + (1^3+2^3) + (2^3+1^3) + (2^3+2^3) = 36.
%p A099903 A099903:=n->(n^5+2*n^4+n^3)/2; seq(A099903(n), n=1..30); # _Wesley Ivan Hurt_, Feb 26 2014
%t A099903 Table[(n^5+2*n^4+n^3)/2, {n, 30}] (* _Wesley Ivan Hurt_, Feb 26 2014 *)
%o A099903 (PARI) a(n)=(n^5+2*n^4+n^3)/2 \\ _Charles R Greathouse IV_, Oct 27 2011
%o A099903 (Magma) [(n^5+2*n^4+n^3)/2: n in [1..30]]; // _Wesley Ivan Hurt_, May 25 2014
%Y A099903 Cf. A099904, A019584, A098077.
%K A099903 nonn,easy
%O A099903 1,1
%A A099903 _Alexander Adamchuk_, Oct 29 2004