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A066967 Total sum of odd parts in all partitions of n. 9
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

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

LINKS

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

FORMULA

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

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);

# 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-Fran├žois Alcover, Jan 24 2014, after Emeric Deutsch *)

Map[Total[Select[Flatten[IntegerPartitions[#]], OddQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

CROSSREFS

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

Sequence in context: A023855 A191832 A066964 * A222450 A032007 A091295

Adjacent sequences:  A066964 A066965 A066966 * A066968 A066969 A066970

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Jan 26 2002

EXTENSIONS

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

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)