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%I #65 Jan 12 2021 12:57:27
%S 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 96,111,128,148,170,195,224,256,293,334,380,432,491,557,630,713,805,
%U 908,1024,1152,1295,1455,1632,1829,2048,2291,2560,2859,3189,3554,3958,4404
%N Number of partitions of n into distinct parts such that number of parts is odd.
%C Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H Alois P. Heinz, <a href="/A067659/b067659.txt">Table of n, a(n) for n = 0..1000</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="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 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 For g.f. see under A067661.
%F a(n) = (A000009(n)-A010815(n))/2. - _Vladeta Jovovic_, Feb 24 2002
%F 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 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 a(n) = A067661(n) - A010815(n). - _Andrey Zabolotskiy_, Apr 12 2017
%F A000009(n) = a(n) + A067661(n). - _Gus Wiseman_, Jan 09 2021
%e From _Gus Wiseman_, Jan 09 2021: (Start)
%e The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
%e 5 6 7 8 9 A B C D E F
%e 321 421 431 432 532 542 543 643 653 654
%e 521 531 541 632 642 652 743 753
%e 621 631 641 651 742 752 762
%e 721 731 732 751 761 843
%e 821 741 832 842 852
%e 831 841 851 861
%e 921 931 932 942
%e A21 941 951
%e A31 A32
%e B21 A41
%e B31
%e C21
%e 54321
%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, 0):
%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, 0]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jan 16 2015, after _Alois P. Heinz_ *)
%t CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* _Andrey Zabolotskiy_, Apr 12 2017 *)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* _Gus Wiseman_, Jan 09 2021 *)
%o (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 (PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);
%o gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
%o concat( [0], Vec(gf) ) /* _Joerg Arndt_, Oct 20 2012 */
%o (PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
%o gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
%o concat( [0], Vec(gf) ) \\ _Joerg Arndt_, Apr 01 2014
%Y Dominates A000009.
%Y Numbers with these strict partitions as binary indices are A000069.
%Y The non-strict version is A027193.
%Y The Heinz numbers of these partitions are A030059.
%Y The even version is A067661.
%Y The version for rank is A117193, with non-strict version A101707.
%Y The ordered version is A332304, with non-strict version A166444.
%Y Other cases of odd length:
%Y - A024429 counts set partitions of odd length.
%Y - A089677 counts ordered set partitions of odd length.
%Y - A174726 counts ordered factorizations of odd length.
%Y - A339890 counts factorizations of odd length.
%Y A008289 counts strict partitions by sum and length.
%Y A026804 counts partitions whose least part is odd, with strict case A026832.
%Y Cf. A000700, A027187, A030229, A117192, A332305.
%K easy,nonn
%O 0,7
%A _Naohiro Nomoto_, Feb 23 2002
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