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A116680 Number of even parts in all partitions of n into distinct parts. 8

%I

%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.

%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);

%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

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Last modified April 17 06:46 EDT 2021. Contains 343059 sequences. (Running on oeis4.)