OFFSET
1,2
COMMENTS
Partial sums of A206436. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 02 2023: (Start)
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)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).
FORMULA
a(n) = 2*Sum_{k=1..floor(n/2)} sigma(k)*numbpart(n-2*k).
a(n) = Sum_{k=0..n} k*A113686(n,k). - Emeric Deutsch, Feb 20 2006
G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Feb 20 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018
EXAMPLE
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.
MAPLE
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);
# Emeric Deutsch, Feb 20 2006
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012
MATHEMATICA
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 *)
Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)
PROG
(PARI) a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ Joerg Arndt, Jan 24 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Jan 26 2002
EXTENSIONS
More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002
More terms from Emeric Deutsch, Feb 20 2006
STATUS
approved