A194552 Sum of all parts > 1 of all partitions of n.

0, 2, 5, 13, 23, 47, 75, 131, 203, 323, 477, 729, 1041, 1517, 2132, 3012, 4134, 5718, 7713, 10453, 13918, 18538, 24357, 32037, 41612, 54040, 69538, 89362, 113925, 145095, 183473, 231697, 290899, 364577, 454632, 566016, 701436, 867800, 1069430, 1315550, 1612595

Offset: 1

Author

Omar E. Pol, Dec 11 2011

Keywords

• nonn

Comments

Also the total number of missing parts in the partitions of n. A missing part of a partition of n is any number from 1 to n not occurring as a part. For example for n = 3, 1,2 are missing from 3; 3 is missing from 2+1, and 2,3 are missing from 1+1+1, for a total of a(3) = 5. - George Beck, Oct 23 2014

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Partial sums of A138880.

Cf. A096541, A135010, A138121, A138135.

Programs

Maple

b:= proc(n, i) option remember; local h, t;
      if n<0 or i<1 then [0, 0]
    elif n=0 or i=1 then [1, 0]
    else h:= b(n, i-1); t:= b(n-i, i);
         [h[1]+t[1], h[2]+t[2] +t[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50); # Alois P. Heinz, Dec 14 2011

Mathematica

a[n_] := n PartitionsP[n] -Total@Table[PartitionsP[k], {k, 0, n - 1}]; a /@ Range[40] (* George Beck, Oct 23 2014 *)

Formula

a(n) = A066186(n) - A000070(n-1).

a(n) = n * A000041(n) - A000070(n-1). - George Beck, Oct 24 2014

G.f.: (x/(1 - x)) * (d/dx) Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Mar 06 2021