login
Number of partitions of n into distinct parts such that number of parts is even.
53

%I #57 Feb 09 2021 02:52:22

%S 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,

%T 96,111,128,148,170,195,224,256,292,334,380,432,491,556,630,713,805,

%U 908,1024,1152,1295,1455,1632,1829,2049,2291,2560,2859,3189,3554,3959,4404

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

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

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

%H Alois P. Heinz, <a href="/A067661/b067661.txt">Table of n, a(n) for n = 0..10000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, end of section 16.4.2 "Partitions into distinct parts", pp.348ff

%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_e(n).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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]

%F 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}.

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

%F Prod_{k >= 1} [ 1, q^k; q^k, 1 ] = [ A(q), B(q); B(q), A(q) ] [_Bill Gosper_, Jun 25 2005]

%F a(n) = (A000009(n)+A010815(n))/2. - _Vladeta Jovovic_, Feb 24 2002

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

%F a(n) + A067659(n) = A000009(n). - _R. J. Mathar_, Jun 18 2016

%F a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, May 24 2018

%F A000009(n) = a(n) + A067659(n). - _Gus Wiseman_, Jan 09 2021

%F From _Peter Bala_, Feb 05 2021: (Start)

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

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

%F 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)).

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

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

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

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

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

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

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

%e 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 + ...

%e From _Gus Wiseman_, Jan 08 2021: (Start)

%e The a(3) = 1 through a(14) = 11 partitions (A-D = 10..13):

%e 21 31 32 42 43 53 54 64 65 75 76 86

%e 41 51 52 62 63 73 74 84 85 95

%e 61 71 72 82 83 93 94 A4

%e 81 91 92 A2 A3 B3

%e 4321 A1 B1 B2 C2

%e 5321 5421 C1 D1

%e 6321 5431 5432

%e 6421 6431

%e 7321 6521

%e 7421

%e 8321

%e (End)

%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,

%p `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))

%p end:

%p a:= n-> b(n$2, 1):

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 01 2014

%t 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_ *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* _Michael Somos_, May 06 2015 *)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,30}] (* _Gus Wiseman_, Jan 08 2021 *)

%o (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 */

%o (PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);

%o gf=sum(n=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );

%o Vec(gf) \\ _Joerg Arndt_, Apr 01 2014

%Y Dominates A000009.

%Y Numbers with these strict partitions as binary indices are A001969.

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

%Y The Heinz numbers of these partitions are A030229.

%Y The odd version is A067659, ranked by A030059.

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

%Y Other cases of even length:

%Y - A024430 counts set partitions of even length.

%Y - A034008 counts compositions of even length.

%Y - A052841 counts ordered set partitions of even length.

%Y - A174725 counts ordered factorizations of even length.

%Y - A332305 counts strict compositions of even length

%Y - A339846 counts factorizations of even length.

%Y A008289 counts strict partitions by sum and length.

%Y A026805 counts partitions whose least part is even.

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

%K easy,nonn

%O 0,6

%A _Naohiro Nomoto_, Feb 23 2002