A067659 Number of partitions of n into distinct parts such that number of parts is odd.

0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404

Offset: 0

Author

Naohiro Nomoto, Feb 23 2002

Keywords

• easy
• nonn

Comments

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			From _Gus Wiseman_, Jan 09 2021: (Start)
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
  5   6     7     8     9     A     B     C     D     E     F
      321   421   431   432   532   542   543   643   653   654
                  521   531   541   632   642   652   743   753
                        621   631   641   651   742   752   762
                              721   731   732   751   761   843
                                    821   741   832   842   852
                                          831   841   851   861
                                          921   931   932   942
                                                A21   941   951
                                                      A31   A32
                                                      B21   A41
                                                            B31
                                                            C21
                                                            54321
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
• 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 q_o(n).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Dominates A000009.

Numbers with these strict partitions as binary indices are A000069.

The non-strict version is A027193.

The Heinz numbers of these partitions are A030059.

The even version is A067661.

The version for rank is A117193, with non-strict version A101707.

The ordered version is A332304, with non-strict version A166444.

Other cases of odd length:

- A024429 counts set partitions of odd length.

- A089677 counts ordered set partitions of odd length.

- A174726 counts ordered factorizations of odd length.

- A339890 counts factorizations of odd length.

A008289 counts strict partitions by sum and length.

A026804 counts partitions whose least part is odd, with strict case A026832.

Cf. A000700, A027187, A030229, A117192, A332305.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 09 2021 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */

PARI

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
concat( [0], Vec(gf) )  /* Joerg Arndt, Oct 20 2012 */

PARI

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
concat( [0], Vec(gf) )  \\ Joerg Arndt, Apr 01 2014

Formula

For g.f. see under A067661.

a(n) = (A000009(n)-A010815(n))/2. - Vladeta Jovovic, Feb 24 2002

Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006

G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]

a(n) = A067661(n) - A010815(n). - Andrey Zabolotskiy, Apr 12 2017

A000009(n) = a(n) + A067661(n). - Gus Wiseman, Jan 09 2021