A100824 Number of partitions of n with at most one odd part.

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Author

Vladeta Jovovic, Jan 13 2005

Keywords

• easy
• nonn

Comments

From Gus Wiseman, Jan 21 2022: (Start)

Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:

1 11 21 22 32 33 43 44 54

111 1111 221 2211 331 2222 441

2111 111111 2221 3311 3222

11111 3211 221111 3321

22111 11111111 4311

211111 22221

1111111 33111

222111

321111

2211111

21111111

111111111

(End)

Examples

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A008951, A000070, A000097, A000098, A000710.

The case of alternating sum 0 (equality) is A000070.

A multiplicative version is A339846.

These partitions are ranked by A349150, conjugate A349151.

A000041 = integer partitions, strict A000009.

A027187 = partitions of even length, strict A067661, ranked by A028260.

A027193 = partitions of odd length, ranked by A026424.

A058695 = partitions of odd numbers.

A103919 = partitions by sum and alternating sum (reverse: A344612).

A277103 = partitions with the same number of odd parts as their conjugate.

Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

Programs

Maple

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

Mathematica

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

PARI

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

Formula

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).

a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016

a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005