A027193 Number of partitions of n into an odd number of parts.

0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Number of partitions of n in which greatest part is odd.

Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].

Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006

Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities. Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021

In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 2014

Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

Examples

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
From _Gus Wiseman_, Feb 11 2021: (Start)
The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (321)    (331)      (422)
                          (11111)  (411)    (421)      (431)
                                   (21111)  (511)      (521)
                                            (22111)    (611)
                                            (31111)    (22211)
                                            (1111111)  (32111)
                                                       (41111)
                                                       (2111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.
  (1)  (11)  (3)    (31)    (5)      (33)      (7)        (53)
             (111)  (1111)  (32)     (51)      (52)       (71)
                            (311)    (321)     (322)      (332)
                            (11111)  (3111)    (331)      (521)
                                     (111111)  (511)      (3221)
                                               (3211)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (32111)
                                                          (311111)
                                                          (11111111)
(End)
		

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.

Links

• T. D. Noe, Table of n, a(n) for n = 0..999
• Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
• D. R. C. Chaves, Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan, 2011. In Portuguese.
• Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_0(n).

Crossrefs

Cf. A000041, A000700, A000701, A026837, A046682.

The Heinz numbers of these partitions are A026424 or A244991.

The even-length version is A027187.

The case of odd sum as well as length is A160786, ranked by A340931.

The case of odd maximum as well as length is A340385.

Other cases of odd length:

- A024429 counts set partitions of odd length.

- A067659 counts strict partitions of odd length.

- A089677 counts ordered set partitions of odd length.

- A166444 counts compositions of odd length.

- A174726 counts ordered factorizations of odd length.

- A332304 counts strict compositions of odd length.

- A339890 counts factorizations of odd length.

A000009 counts partitions into odd parts, ranked by A066208.

A026804 counts partitions whose least part is odd.

A058695 counts partitions of odd numbers, ranked by A300063.

A072233 counts partitions by sum and length.

A101707 counts partitions of odd positive rank.

Cf. A078408, A174725, A236914, A300272, A340831.

Programs

Maple

g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # Emeric Deutsch, Apr 05 2006

Mathematica

nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Dec 01 2012 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)

PARI

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */

PARI

q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014

Formula

a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.

For g.f. see under A027187.

G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006

G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014

G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014

a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - Reinhard Zumkeller, Apr 22 2006