0,5

Also number of partitions of n with at least one even part. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 10 2003. Example: a(5)=4 because we have [4,1],[3,2],[2,2,1] and [2,1,1,1] ([5],[3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006

Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0],[4,1,0],[3,2,0] and [3,1,1,0] ([2,2,1,0],[2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006

H. Bottomley, Illustration for A000009, A000041, A047967

G.f.=sum(x^(2k)*product(1+x^j, j=k+1..infinity)/product(1-x^j, j=1..k), k=1..infinity). G.f.=sum(x^(2k)/[product(1-x^j, j=1..2k)*product(1-x^(2j+1), j=k..infinity)], k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006

a(5)=4 because we have [3,1,1],[2,2,1],[2,1,1,1] and [1,1,1,1,1] ([5],[4,1] and [3,2] do not qualify).

g:=sum(x^(2*k)*product(1+x^j, j=k+1..70)/product(1-x^j, j=1..k), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006

Clear[fQ, fP, lst, n]; fQ[n_]:=PartitionsQ[n]; fP[n_]:=PartitionsP[n]; lst={}; Do[AppendTo[lst, fP[n]-fQ[n]], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 19 2009]

A000041(n)-A000009(n).

Cf. A038348.

Sequence in context: A047625 A147871 A004397 this_sequence A147955 A134591 A058611

Adjacent sequences: A047964 A047965 A047966 this_sequence A047968 A047969 A047970

nonn

N. J. A. Sloane (njas(AT)research.att.com).