A066897 - OEIS

%I #82 Jan 15 2025 04:17:39

%S 1,2,5,8,15,24,39,58,90,130,190,268,379,522,722,974,1317,1754,2330,

%T 3058,4010,5200,6731,8642,11068,14076,17864,22528,28347,35490,44320,

%U 55100,68355,84450,104111,127898,156779,191574,233625,284070,344745,417292,504151

%N Total number of odd parts in all partitions of n.

%C 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

%C Column 1 of A206563. - _Omar E. Pol_, Feb 15 2012

%C 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

%C From _Omar E. Pol_, Apr 02 2023: (Start)

%C Convolution of A000041 and A001227.

%C Convolution of A002865 and A060831.

%C 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)

%C a(n) is odd iff n is a term of A067567 (proof: n*p(n) = the sum of the parts in all the partitions of n == the number of odd parts in all partitions of n (mod 2). Hence the number of odd parts in all partitions of n is odd iff n*p(n) is odd, equivalently, iff both n and p(n) are odd). - _Peter Bala_, Jan 11 2025

%H Vaclav Kotesovec, <a href="/A066897/b066897.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%F 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

%F a(n) = Sum_{k=0..n} k*A103919(n,k). - _Emeric Deutsch_, Mar 13 2006

%F G.f.: Sum_{j>=1}(x^(2j-1)/(1-x^(2j-1)))/Product_{j>=1}(1-x^j). - _Emeric Deutsch_, Mar 13 2006

%F a(n) = A066898(n) + A209423(n) = A006128(n) - A066898(n). [_Reinhard Zumkeller_, Mar 09 2012]

%F a(n) = A207381(n) - A207382(n). - _Omar E. Pol_, Mar 11 2012

%F a(n) = (A006128(n) + A209423(n))/2. - _Vaclav Kotesovec_, May 25 2018

%F 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

%e 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.

%p 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);

%p # _Emeric Deutsch_, Mar 13 2006

%p b:= proc(n, i) option remember; local f, g;

%p       if n=0 or i=1 then [1, n]

%p     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));

%p          [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]

%p       fi

%p     end:

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

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

%p # _Alois P. Heinz_, Mar 22 2012

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

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

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

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

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

%t %% - %                   (* A209423 *)

%t (* _Clark Kimberling_, Mar 08 2012 *)

%t 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_ *)

%o (Haskell)

%o a066897 = p 0 1 where

%o    p o _             0 = o

%o    p o k m | m < k     = 0

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

%o -- _Reinhard Zumkeller_, Mar 09 2012

%o (Haskell)

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

%o    ps _ 0 = [[]]

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

%o -- _Reinhard Zumkeller_, Jul 13 2013

%Y Cf. A000041, A001227, A001620, A002865, A006128, A060831, A066898, A066966, A066967, A103919, A206563, A207381, A207382, A209423, A338156.

%K easy,nonn,changed

%O 1,2

%A _Naohiro Nomoto_, Jan 24 2002

%E More terms from _Vladeta Jovovic_, Jan 26 2002