A052839 Number of partitions of n into distinct summands (A000009), plus 1 (apart from the first term).

1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 28, 33, 39, 47, 55, 65, 77, 90, 105, 123, 143, 166, 193, 223, 257, 297, 341, 391, 449, 513, 586, 669, 761, 865, 983, 1114, 1261, 1427, 1611, 1817, 2049, 2305, 2591, 2911, 3265, 3659, 4098, 4583, 5121, 5719, 6379

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 806

Crossrefs

Apart from the first term equals A000009 + 1 and also the left edge of A068049.

Programs

Maple

spec := [S,{C=Sequence(Z,1 <= card),B=PowerSet(C),S=Union(B,C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..67);
Or: with(gfun,seriestolist); seriestolist(series(mul(1+z^i,i=1..81)+z/(1-z),z,81));

Mathematica

a[n_] := If[n == 0, 1, PartitionsQ[n] + 1];
a /@ Range[0, 55] (* Jean-François Alcover, May 07 2020 *)

Formula

G.f.: (-x-exp(Sum(-(-1)^(j[1]+1)*x^j[1]/(x^j[1]-1)/j[1], j[1]=1 .. infinity))+exp(Sum(-(-1)^(j[1]+1)*x^j[1]/(x^j[1]-1)/j[1], j[1]=1 .. infinity))*x)/(-1+x)

Extensions

Edited by Antti Karttunen, Feb 13 2002, based on information received from Bruno Salvy.