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%I A067659 #66 Feb 16 2025 08:32:45
%S A067659 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,
%T A067659 96,111,128,148,170,195,224,256,293,334,380,432,491,557,630,713,805,
%U A067659 908,1024,1152,1295,1455,1632,1829,2048,2291,2560,2859,3189,3554,3958,4404
%N A067659 Number of partitions of n into distinct parts such that number of parts is odd.
%C A067659 Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H A067659 Alois P. Heinz, <a href="/A067659/b067659.txt">Table of n, a(n) for n = 0..1000</a>
%H A067659 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 A067659 Mircea Merca, <a href="https://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_o(n).
%H A067659 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A067659 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A067659 For g.f. see under A067661.
%F A067659 a(n) = (A000009(n)-A010815(n))/2. - _Vladeta Jovovic_, Feb 24 2002
%F A067659 Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - _Michael Somos_, Feb 14 2006
%F A067659 G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [_Joerg Arndt_, Apr 01 2014]
%F A067659 a(n) = A067661(n) - A010815(n). - _Andrey Zabolotskiy_, Apr 12 2017
%F A067659 A000009(n) = a(n) + A067661(n). - _Gus Wiseman_, Jan 09 2021
%e A067659 From _Gus Wiseman_, Jan 09 2021: (Start)
%e A067659 The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
%e A067659 5 6 7 8 9 A B C D E F
%e A067659 321 421 431 432 532 542 543 643 653 654
%e A067659 521 531 541 632 642 652 743 753
%e A067659 621 631 641 651 742 752 762
%e A067659 721 731 732 751 761 843
%e A067659 821 741 832 842 852
%e A067659 831 841 851 861
%e A067659 921 931 932 942
%e A067659 A21 941 951
%e A067659 A31 A32
%e A067659 B21 A41
%e A067659 B31
%e A067659 C21
%e A067659 54321
%e A067659 (End)
%p A067659 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A067659 `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
%p A067659 end:
%p A067659 a:= n-> b(n$2, 0):
%p A067659 seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 01 2014
%t A067659 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_ *)
%t A067659 CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* _Andrey Zabolotskiy_, Apr 12 2017 *)
%t A067659 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* _Gus Wiseman_, Jan 09 2021 *)
%o A067659 (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 */
%o A067659 (PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);
%o A067659 gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
%o A067659 concat( [0], Vec(gf) ) /* _Joerg Arndt_, Oct 20 2012 */
%o A067659 (PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
%o A067659 gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
%o A067659 concat( [0], Vec(gf) ) \\ _Joerg Arndt_, Apr 01 2014
%Y A067659 Dominates A000009.
%Y A067659 Numbers with these strict partitions as binary indices are A000069.
%Y A067659 The non-strict version is A027193.
%Y A067659 The Heinz numbers of these partitions are A030059.
%Y A067659 The even version is A067661.
%Y A067659 The version for rank is A117193, with non-strict version A101707.
%Y A067659 The ordered version is A332304, with non-strict version A166444.
%Y A067659 Other cases of odd length:
%Y A067659 - A024429 counts set partitions of odd length.
%Y A067659 - A089677 counts ordered set partitions of odd length.
%Y A067659 - A174726 counts ordered factorizations of odd length.
%Y A067659 - A339890 counts factorizations of odd length.
%Y A067659 A008289 counts strict partitions by sum and length.
%Y A067659 A026804 counts partitions whose least part is odd, with strict case A026832.
%Y A067659 Cf. A000700, A027187, A030229, A117192, A332305.
%K A067659 easy,nonn
%O A067659 0,7
%A A067659 _Naohiro Nomoto_, Feb 23 2002