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

0,4

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)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)

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

Example

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

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
# Alternative:
a:= n-> (p-> p(n)-`if`(n::odd, 0, p(n/2)))(combinat[numbpart]):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 25 2026

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 *)

Prog

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

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.

Sequence in context: A117989 A391794 A241642 * A281616 A110618 A395924

Adjacent sequences: A086540 A086541 A086542 * A086544 A086545 A086546

Keyword

nonn

Author

Vladeta Jovovic, Sep 10 2003

Status

approved