A066967 Total sum of odd parts in all partitions of n.

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

Offset: 1

Author

Vladeta Jovovic, Jan 26 2002

Keywords

• nonn

Comments

Partial sums of A206435. - Omar E. Pol, Mar 17 2012

From Omar E. Pol, Apr 01 2023: (Start)

Convolution of A000041 and A000593.

Convolution of A002865 and A078471.

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)

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
• 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).

Crossrefs

Cf. A000041, A000593, A066897, A066898, A113685, A206435.

Cf. A002865, A078471.

Programs

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);
# Emeric Deutsch, Feb 19 2006
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);
# Alois P. Heinz, Mar 22 2012

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 *)

Formula

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.

a(n) = sum(k*A113685(n,k), k=0..n). - Emeric Deutsch, Feb 19 2006

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

a(n) = A066186(n) - A066966(n). - Omar E. Pol, Mar 10 2012

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002