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%I #54 Dec 12 2023 08:18:27
%S 0,2,2,10,12,30,40,82,110,190,260,422,570,860,1160,1690,2252,3170,
%T 4190,5760,7540,10142,13164,17450,22442,29300,37410,48282,61170,78132,
%U 98310,124444,155582,195310,242722,302570,373882,462954,569130,700570,856970
%N Total sum of even parts in all partitions of n.
%C Partial sums of A206436. - _Omar E. Pol_, Mar 17 2012
%C From _Omar E. Pol_, Apr 02 2023: (Start)
%C Convolution of A000041 and A146076.
%C Convolution of A002865 and A271342.
%C a(n) is also the sum of all even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)
%H Alois P. Heinz, <a href="/A066966/b066966.txt">Table of n, a(n) for n = 1..1000</a>
%H George E. Andrews and Mircea Merca, <a href="https://doi.org/10.1016/j.jcta.2023.105849">A further look at the sum of the parts with the same parity in the partitions of n</a>, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).
%F a(n) = 2*Sum_{k=1..floor{n/2)} sigma(k)*numbpart(n-2*k).
%F a(n) = Sum_{k=0..n} k*A113686(n,k). - _Emeric Deutsch_, Feb 20 2006
%F G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - _Emeric Deutsch_, Feb 20 2006
%F a(n) = A066186(n) - A066967(n). - _Omar E. Pol_, Mar 10 2012
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - _Vaclav Kotesovec_, May 29 2018
%e a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the even parts is 4+2+2+2 = 10.
%p g:=sum(2*j*x^(2*j)/(1-x^(2*j)),j=1..55)/product(1-x^j,j=1..55): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..41);
%p # _Emeric Deutsch_, Feb 20 2006
%p b:= proc(n, i) option remember; local f, g;
%p if n=0 or i=1 then [1, 0]
%p else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=1..50);
%p # _Alois P. Heinz_, Mar 22 2012
%t max = 50; g = Sum[2*j*x^(2*j)/(1 - x^(2*j)), {j, 1, max}]/Product[1 - x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max - 1}] (* _Jean-François Alcover_, Jan 24 2014, after _Emeric Deutsch_ *)
%t Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* _Peter J. C. Moses_, Mar 14 2014 *)
%o (PARI) a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ _Joerg Arndt_, Jan 24 2014
%Y Cf. A000041, A000203, A002865, A066186, A066897, A066898, A066967, A113686, A146076, A206436, A271342, A338156.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Jan 26 2002
%E More terms from _Naohiro Nomoto_ and _Sascha Kurz_, Feb 07 2002
%E More terms from _Emeric Deutsch_, Feb 20 2006
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