A241205 - OEIS

%I #18 Aug 16 2017 08:45:31

%S 0,0,1,14,95,436,1554,4632,12078,28380,61347,123838,236093,428792,

%T 746980,1255008,2042652,3232584,4989381,7530270,11137819,16174796,

%U 23101430,32495320,45074250,61722180,83518695,111772206,148057209,194255920,252604616,325745024,416781112,529341648,667648905

%N Sum of x*y^2*z^3 for positive integers x,y,z with x + y + z = n.

%H Robert Israel, <a href="/A241205/b241205.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n^2*(n-1)*(n-2)*(n+2)*(n+1)*(3*n^2+1)/10080. - corrected by _Eric Rowland_, Aug 15 2017

%F G.f: x^3*(1+x)*(1+4*x+x^2)/(1-x)^9.

%e For n=3 the only positive integer solution to x+y+z=3 is (1,1,1) so a(3) = 1.

%e For n=4 the solutions are (1,1,2),(1,2,1),(2,1,1) so a(4) = 2^3 + 2^2 + 2 = 14.

%p A241205:= n -> n^2*(n-1)*(n-2)*(n+2)*(n+1)*(3*n^2+1)/10080; # _Robert Israel_, Apr 17 2014

%t Rest[CoefficientList[Series[x^3 (1 + x) (1 + 4 x + x^2)/(1 - x)^9, {x, 0, 35}], x]] (* _Vincenzo Librandi_, Jul 01 2014 *)

%K nonn,easy

%O 1,4

%A _Robert Israel_, Apr 17 2014