login
Number of partitions of n into distinct summands (A000009), plus 1 (apart from the first term).
3

%I #13 May 07 2020 06:47:55

%S 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,

%T 143,166,193,223,257,297,341,391,449,513,586,669,761,865,983,1114,

%U 1261,1427,1611,1817,2049,2305,2591,2911,3265,3659,4098,4583,5121,5719,6379

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

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=806">Encyclopedia of Combinatorial Structures 806</a>

%F 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)

%p spec := [S,{C=Sequence(Z,1 <= card),B=PowerSet(C),S=Union(B,C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..67);

%p Or: with(gfun,seriestolist); seriestolist(series(mul(1+z^i,i=1..81)+z/(1-z),z,81));

%t a[n_] := If[n == 0, 1, PartitionsQ[n] + 1];

%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 07 2020 *)

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

%K easy,nonn

%O 0,2

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

%E Edited by _Antti Karttunen_, Feb 13 2002, based on information received from Bruno Salvy.