A086543 Number of partitions of n with at least one odd part.

0, 1, 1, 3, 3, 7, 8, 15, 17, 30, 35, 56, 66, 101, 120, 176, 209, 297, 355, 490, 585, 792, 946, 1255, 1498, 1958, 2335, 3010, 3583, 4565, 5428, 6842, 8118, 10143, 12013, 14883, 17592, 21637, 25525, 31185, 36711, 44583, 52382, 63261, 74173, 89134, 104303, 124754, 145698, 173525, 202268

Offset: 0

Author

Vladeta Jovovic, Sep 10 2003

Keywords

• nonn

Comments

From Gus Wiseman, Oct 12 2023: (Start)

Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:

(1) (2) (3) (4) (5) (6) (7)

(21) (31) (32) (42) (43)

(111) (1111) (41) (51) (52)

(221) (222) (61)

(311) (411) (322)

(2111) (2211) (331)

(11111) (21111) (421)

(111111) (511)

(2221)

(3211)

(4111)

(22111)

(31111)

(211111)

(1111111)

Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:

(1) (11) (3) (31) (5) (33) (7)

(21) (211) (32) (51) (43)

(111) (1111) (41) (321) (52)

(221) (411) (61)

(311) (2211) (322)

(2111) (3111) (331)

(11111) (21111) (421)

(111111) (511)

(2221)

(3211)

(4111)

(22111)

(31111)

(211111)

(1111111)

(End)

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A038348, A047967.

The complement is counted by A035363, ranks A344415.

These partitions have ranks A366322.

A025065 counts partitions with sum <= twice length, ranks A344296.

A110618 counts partitions with sum >= twice maximum, ranks A344291.

Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.

Programs

Maple

g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # Emeric Deutsch, Mar 30 2006

Mathematica

nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)
Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* Gus Wiseman, Oct 12 2023 *)

PARI

x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ Joerg Arndt, May 04 2013

Formula

A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).

G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - Emeric Deutsch, Mar 30 2006

G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - Joerg Arndt, May 04 2013