A066897 - OEIS

A066897

Total number of odd parts in all partitions of n.

1, 2, 5, 8, 15, 24, 39, 58, 90, 130, 190, 268, 379, 522, 722, 974, 1317, 1754, 2330, 3058, 4010, 5200, 6731, 8642, 11068, 14076, 17864, 22528, 28347, 35490, 44320, 55100, 68355, 84450, 104111, 127898, 156779, 191574, 233625, 284070, 344745, 417292, 504151

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OFFSET

1,2

COMMENTS

Also sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n (Cf. A206563). - Omar E. Pol, Feb 12 2012

Column 1 of A206563. - Omar E. Pol, Feb 15 2012

Suppose that p=[p(1),p(2),p(3),...] is a partition of n with parts in nonincreasing order. Let f(p) = p(1) - p(2) + p(3) - ... be the alternating sum of parts of p and let F(n) = sum of alternating sums of all partitions of n. Conjecture: F(n) = A066897(n) for n >= 1. - Clark Kimberling, May 17 2019

From Omar E. Pol, Apr 02 2023: (Start)

Convolution of A000041 and A001227.

Convolution of A002865 and A060831.

a(n) is also the total number of 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

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

FORMULA

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A001227(k)=number of odd divisors of k and numbpart() is A000041. - Vladeta Jovovic, Jan 26 2002

a(n) = Sum_{k=0..n} k*A103919(n,k). - Emeric Deutsch, Mar 13 2006

G.f.: Sum_{j>=1}(x^(2j-1)/(1-x^(2j-1)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Mar 13 2006

a(n) = A066898(n) + A209423(n) = A006128(n) - A066898(n). [Reinhard Zumkeller, Mar 09 2012]

a(n) = A207381(n) - A207382(n). - Omar E. Pol, Mar 11 2012

a(n) = (A006128(n) + A209423(n))/2. - Vaclav Kotesovec, May 25 2018

a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(24*n/Pi^2)) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

EXAMPLE

a(4) = 8 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], we have a total of 0+2+0+2+4=8 odd parts.

MAPLE

g:=sum(x^(2*j-1)/(1-x^(2*j-1)), j=1..70)/product(1-x^j, j=1..70): gser:=series(g, x=0, 45): seq(coeff(gser, x^n), n=1..44);

# Emeric Deutsch, Mar 13 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]]

end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..50);

# Alois P. Heinz, Mar 22 2012

MATHEMATICA

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]

o[n_] := Sum[f[n, i], {i, 1, n, 2}]

e[n_] := Sum[f[n, i], {i, 2, n, 2}]

Table[o[n], {n, 1, 45}] (* A066897 *)

Table[e[n], {n, 1, 45}] (* A066898 *)

%% - % (* A209423 *)

(* Clark Kimberling, Mar 08 2012 *)

b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 26 2015, after Alois P. Heinz *)

PROG

(Haskell)

a066897 = p 0 1 where

p o _ 0 = o

p o k m | m < k = 0

| otherwise = p (o + mod k 2) k (m - k) + p o (k + 1) m

-- Reinhard Zumkeller, Mar 09 2012

(Haskell)

a066897 = length . filter odd . concat . ps 1 where

ps _ 0 = [[]]

ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]

-- Reinhard Zumkeller, Jul 13 2013