A067661 Number of partitions of n into distinct parts such that number of parts is even.

1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404

Offset: 0

Author

Naohiro Nomoto, Feb 23 2002

Keywords

• easy
• nonn

Comments

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

Examples

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(3) = 1 through a(14) = 11 partitions (A-D = 10..13):
  21   31   32   42   43   53   54   64     65     75     76     86
            41   51   52   62   63   73     74     84     85     95
                      61   71   72   82     83     93     94     A4
                                81   91     92     A2     A3     B3
                                     4321   A1     B1     B2     C2
                                            5321   5421   C1     D1
                                                   6321   5431   5432
                                                          6421   6431
                                                          7321   6521
                                                                 7421
                                                                 8321
(End)
		

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• 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_e(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 A001969.

The non-strict case is A027187, ranked by A028260.

The Heinz numbers of these partitions are A030229.

The odd version is A067659, ranked by A030059.

The version for rank is A117192, with positive case A101708.

Other cases of even length:

- A024430 counts set partitions of even length.

- A034008 counts compositions of even length.

- A052841 counts ordered set partitions of even length.

- A174725 counts ordered factorizations of even length.

- A332305 counts strict compositions of even length

- A339846 counts factorizations of even length.

A008289 counts strict partitions by sum and length.

A026805 counts partitions whose least part is even.

Cf. A000700, A027193, A058696, A072233, A096373, A236913, A244990, A300061.

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, 1):
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, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* Michael Somos, May 06 2015 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 08 2021 *)

PARI

{a(n) = my(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=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

Formula

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q){2n} [_Bill Gosper, Jun 25 2005]

Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}.

Also we have the following identity involving 2 X 2 matrices:

Prod_{k >= 1} [ 1, q^k; q^k, 1 ] = [ A(q), B(q); B(q), A(q) ] [Bill Gosper, Jun 25 2005]

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

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

a(n) + A067659(n) = A000009(n). - R. J. Mathar, Jun 18 2016

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018

A000009(n) = a(n) + A067659(n). - Gus Wiseman, Jan 09 2021

From Peter Bala, Feb 05 2021: (Start)

G.f.: A(x) = (1/2)*((Product_{n >= 0} 1 + x^n) + (Product_{n >= 0} 1 - x^n)).

Let B(x) denote the g.f. of A067659. Then

A(x)^2 - B(x)^2 = A(x^2) - B(x^2) = Product_{n >= 1} 1 - x^(2*n) = Sum_{n in Z} (-1)^n*x^(n*(3*n+1)).

A(x) + B(x) is the g.f. of A000009.

1/(A(x) - B(x)) is the g.f. of A000041.

(A(x) + B(x))/(A(x) - B(x)) is the g.f. of A015128.

A(x)/(A(x) + B(x)) = Sum_{n >= 0} (-1)^n*x^n^2 = (1 + theta_3(-x))/2.

B(x)/(A(x) - B(x)) is the g.f. of A014968.

A(x)/(A(x^2) - B(x^2)) is the g.f. of A027187.

B(x)/(A(x^2) - B(x^2)) is the g.f. of A027193. (End)