A116676 Number of odd parts in all partitions of n into distinct parts.

0, 1, 0, 2, 2, 3, 4, 5, 8, 10, 14, 16, 22, 26, 34, 43, 54, 64, 80, 96, 116, 142, 170, 202, 242, 288, 340, 404, 474, 556, 652, 762, 886, 1034, 1198, 1389, 1606, 1852, 2132, 2454, 2814, 3224, 3690, 4214, 4804, 5478, 6228, 7072, 8028, 9094, 10290, 11635, 13134

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn

Comments

a(n) = Sum(k*A116675(n,k), k>=0).

Examples

			a(9) = 10 because in the partitions of 9 into distinct parts, namely, [9], [81], [72], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2], we have a total of 10 odd parts.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A116675, A305123, A305124.

Cf. A305082, A015723, A090867, A116680, A067588.

Programs

Maple

f:=product(1+x^j,j=1..64)*sum(x^(2*j-1)/(1+x^(2*j-1)),j=1..35): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..56);
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0] elif i<1 then [0, 0]
    else f:=b(n, i-1); g:=`if`(i>n, [0, 0], b(n-i, min(n-i, i-1)));
         [f[1]+g[1], f[2]+g[2] +irem(i, 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

Mathematica

b[n_, i_] := b[n, i] = Module[{f, g}, Which [n == 0, {1, 0}, i<1 , {0, 0}, True, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, Min[n-i, i-1]]]; {f[[1]] + g[[1]],       f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 0, 60}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

Formula

G.f.: product(1+x^j, j=1..infinity)*sum(x^(2j-1)/(1+x^(2j-1)), j=1..infinity).

For n > 0, a(n) = A015723(n) - A116680(n). - Vaclav Kotesovec, May 26 2018

a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018