|
|
A066967
|
|
Total sum of odd parts in all partitions of n.
|
|
13
|
|
|
1, 2, 7, 10, 23, 36, 65, 94, 160, 230, 356, 502, 743, 1030, 1480, 2006, 2797, 3760, 5120, 6780, 9092, 11902, 15701, 20350, 26508, 34036, 43860, 55822, 71215, 89988, 113792, 142724, 179137, 223230, 278183, 344602, 426687, 525616, 647085, 792950
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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)
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.
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
|
|
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 odd parts is (3+1)+(1+1)+(1+1+1+1) = 10.
|
|
MAPLE
|
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);
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, n]
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 mod 2)*g[1]*i]
fi
end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);
|
|
MATHEMATICA
|
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 *)
Map[Total[Select[Flatten[IntegerPartitions[#]], OddQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|