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%I #25 May 27 2018 10:19:51
%S 0,0,0,2,2,4,4,6,6,8,12,14,18,24,32,38,50,60,76,90,110,134,162,190,
%T 228,270,322,380,446,524,616,720,838,980,1134,1314,1526,1760,2026,
%U 2336,2676,3072,3518,4020,4586,5232,5948,6760,7676,8698,9846,11142,12578
%N Number of parts in all partitions of n into even number of distinct parts.
%H Alois P. Heinz, <a href="/A238132/b238132.txt">Table of n, a(n) for n = 0..5000</a>
%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_e(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))}=A015723(n)-A238131(n).
%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)=6 because the partitions of 8 into even number of distinct parts are: 7+1, 6+2 and 5+3.
%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)[3]:
%p seq(a(n), n=0..60); # _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.
%K nonn
%O 0,4
%A _Mircea Merca_, Feb 18 2014
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