%I #32 May 27 2018 10:20:09
%S 0,1,1,1,1,1,4,4,7,10,13,16,22,25,31,42,48,59,73,89,108,132,156,190,
%T 227,271,318,380,449,526,618,722,841,980,1138,1321,1526,1760,2028,
%U 2333,2683,3070,3517,4017,4584,5228,5948,6757,7673,8696,9845,11132,12577
%N Number of parts in all partitions of n into odd number of distinct parts.
%H Vaclav Kotesovec, <a href="/A238131/b238131.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)
%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 s_o(n).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PolygammaFunction.html">q-Polygamma Function</a>, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F a(n) = (1/2)*A015723(n)+(1/2)*sum{k=0..A235963(n)-1, (-1)^A110654(k)*A000005(n-A001318(k))}.
%F G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) + (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).
%F G.f.: (2 * (x; x)_inf * (log(1-x) + psi_x(1)) - (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov_, Nov 21 2016
%F a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, May 27 2018
%e a(8)=7 because the partitions of 8 into odd number of distinct parts are: 8, 5+2+1 and 4+3+1.
%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, [1, 0$3], b(n, i-1)+`if`(i>n, 0, (p->
%p [p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
%p end:
%p a:= n-> b(n$2)[4]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Dec 27 2015
%t max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] + (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* _Jean-François Alcover_, Dec 27 2015 *)
%Y Cf. A000005, A001318, A015723, A110654, A235963, A079499.
%K nonn
%O 0,7
%A _Mircea Merca_, Feb 18 2014