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

1,2

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

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

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 *)

Crossrefs

Partial sums of A138880.

Cf. A096541, A135010, A138121, A138135.

Sequence in context: A281906 A256491 A106009 * A079780 A178621 A362105

Adjacent sequences: A194549 A194550 A194551 * A194553 A194554 A194555

Keyword

nonn

Author

Omar E. Pol, Dec 11 2011

Status

approved