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A116676 Number of odd parts in all partitions of n into distinct parts. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, May 22 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A116675, A305123, A305124.

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

Sequence in context: A211228 A186505 A228693 * A240575 A176538 A285261

Adjacent sequences:  A116673 A116674 A116675 * A116677 A116678 A116679

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 22 2006

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)