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%I #54 Dec 12 2023 08:18:39
%S 1,2,7,10,23,36,65,94,160,230,356,502,743,1030,1480,2006,2797,3760,
%T 5120,6780,9092,11902,15701,20350,26508,34036,43860,55822,71215,89988,
%U 113792,142724,179137,223230,278183,344602,426687,525616,647085,792950
%N Total sum of odd parts in all partitions of n.
%C Partial sums of A206435. - _Omar E. Pol_, Mar 17 2012
%C From _Omar E. Pol_, Apr 01 2023: (Start)
%C Convolution of A000041 and A000593.
%C Convolution of A002865 and A078471.
%C a(n) is also the sum of all odd 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 odd divisors are also all odd parts of all partitions of n. (End)
%H Vaclav Kotesovec, <a href="/A066967/b066967.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%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) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.
%F a(n) = sum(k*A113685(n,k), k=0..n). - _Emeric Deutsch_, Feb 19 2006
%F G.f.: sum((2i-1)x^(2i-1)/(1-x^(2i-1)), i=1..infinity)/product(1-x^j, j=1..infinity). - _Emeric Deutsch_, Feb 19 2006
%F a(n) = A066186(n) - A066966(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 odd parts is (3+1)+(1+1)+(1+1+1+1) = 10.
%p g:=sum((2*i-1)*x^(2*i-1)/(1-x^(2*i-1)),i=1..50)/product(1-x^j,j=1..50): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..47);
%p # _Emeric Deutsch_, Feb 19 2006
%p b:= proc(n, i) option remember; local f, g;
%p if n=0 or i=1 then [1, n]
%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 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*i-1)*x^(2*i-1)/(1-x^(2*i-1)), {i, 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[#]], OddQ]] &, Range[30]] (* _Peter J. C. Moses_, Mar 14 2014 *)
%Y Cf. A000041, A000593, A066897, A066898, A113685, A206435.
%Y Cf. A002865, A078471.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Jan 26 2002
%E More terms from _Naohiro Nomoto_ and _Sascha Kurz_, Feb 07 2002
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