A220486 - OEIS

%I #13 Feb 21 2013 15:35:36

%S 0,0,3,8,25,42,91,144,243,380,594,852,1287,1834,2580,3616,5015,6822,

%T 9272,12420,16548,21956,28819,37608,48875,63232,81162,103936,132327,

%U 167880,212040,266976,334587,418404,520765,646848,800495,988418,1216059,1493200

%N a(n) = n(p(n)-d(n)): sum of all of parts of all partitions of n with at least one distinct part.

%F a(n) = n*(A000041(n) - A000005(n)) = A066186(n) - A038040(n) =  n*A144300(n).

%e For n = 6

%e -----------------------------------------------------

%e Partitions of 6            Value

%e -----------------------------------------------------

%e 6 .......................... 0  (all parts are equal)

%e 5 + 1 ...................... 6

%e 4 + 2 ...................... 6

%e 4 + 1 + 1 .................. 6

%e 3 + 3 ...................... 0  (all parts are equal)

%e 3 + 2 + 1 .................. 6

%e 3 + 1 + 1 + 1 .............. 6

%e 2 + 2 + 2 .................. 0  (all parts are equal)

%e 2 + 2 + 1 + 1 .............. 6

%e 2 + 1 + 1 + 1 + 1 .......... 6

%e 1 + 1 + 1 + 1 + 1 + 1 ...... 0  (all parts are equal)

%e -----------------------------------------------------

%e The sum of the values is    42

%e On the other hand p(6) = A000041(6) = 11 and d(6) = A000005(6) = 4, so a(6) = 6*(p(6) - d(6)) = 6*(11 - 4) = 6*7 = 42.

%Y Cf. A000005, A000041, A038040, A066186, A144300, A220477.

%K nonn,easy

%O 1,3

%A _Omar E. Pol_, Jan 18 2013