%I #37 Sep 08 2022 08:45:24
%S 0,0,1,1,1,2,4,5,5,8,11,14,18,23,29,37,44,55,69,83,102,124,148,178,
%T 213,253,300,356,421,494,582,680,793,926,1074,1246,1446,1668,1922,
%U 2215,2545,2918,3345,3823,4366,4982,5668,6445,7321,8300,9401,10639,12021,13566
%N Number of even parts in all partitions of n into distinct parts.
%H Vaclav Kotesovec, <a href="/A116680/b116680.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Herden, M. R. Sepanski, J. Stanfill, C. C. Hammon, J. Henningsen, H. Ickes, J. M. Menendez, T. Poe, I. Ruiz, and E. L. Smith, <a href="https://arxiv.org/abs/2010.02788">Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k</a>, arXiv:2010.02788 [math.CO], 2020. See also <a href="http://math.colgate.edu/~integers/w49/w49.pdf">Integers</a> (2022) Vol. 22, #A49.
%H Runqiao Li, Andrew Y. Z. Wang, <a href="https://doi.org/10.1007/s11139-020-00356-w">On the combinatorics of the number of even parts in all partitions with distinct parts</a>, The Raman. J. 56 (2021) 712-727
%F a(n) = Sum_{k >= 0} k*A116679(n,k).
%F G.f.: Product_{j >= 1} (1+x^j) * Sum_{k >= 1} (x^(2*k)/(1+x^(2*k)).
%F For n > 0, a(n) = A015723(n) - A116676(n). - _Vaclav Kotesovec_, May 26 2018
%F a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, May 26 2018
%e a(9)=8 because in the partitions of 9 into distinct parts, namely, [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1], and [4,3,2], we have a total of 8 even parts. [edited by _Rishi Advani_, Jun 07 2019]
%p f:=product(1+x^j,j=1..70)*sum(x^(2*j)/(1+x^(2*j)),j=1..40): fser:=series(f,x=0,65): seq(coeff(fser,x,n),n=0..60);
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, [1, 0],
%p b(n, i-1)+(p-> p+`if`(i::odd, 0, [0, p[1]]))(b(n-i, min(n-i, i-1)))))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, May 24 2022
%t With[{m = 25}, CoefficientList[Series[Product[1+x^j, {j,1,4*m}]* Sum[x^(2*k)/(1+x^(2*k)), {k,1,2*m}], {x,0,3*m}], x]] (* _G. C. Greubel_, Jun 07 2019 *)
%o (PARI) my(m=25); my(x='x+O('x^(3*m))); concat([0, 0], Vec( prod(j=1, 4*m, 1+x^j)*sum(k=1, 2*m, x^(2*k)/(1+x^(2*k))) )) \\ _G. C. Greubel_, Jun 07 2019
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), 3*m); [0,0] cat Coefficients(R!( (&*[1+x^j: j in [1..4*m]])*(&+[x^(2*k)/(1+x^(2*k)): k in [1..2*m]]) )); // _G. C. Greubel_, Jun 07 2019
%o (Sage)
%o m = 25
%o R = PowerSeriesRing(ZZ, 'x')
%o x = R.gen().O(3*m)
%o s = product(1+x^j for j in (1..4*m))*sum(x^(2*k)/(1+x^(2*k)) for k in (1..2*m))
%o [0, 0] + s.coefficients() # _G. C. Greubel_, Jun 07 2019
%Y Cf. A116679, A305121, A305122.
%Y Cf. A305082, A015723, A090867, A067588, A116676.
%K nonn
%O 0,6
%A _Emeric Deutsch_, Feb 22 2006