1,3

a(n) = number of parts of odd multiplicity (each counted only once) in all partitions of n. Example: a(5) = 10 because we have [5'],[4',1'],[3',2'], [3',1,1],[2,2,1'],[2',1',1,1], and [1',1,1,1,1] (the 10 counted parts are marked). - Emeric Deutsch, Feb 08 2016

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

a(n) = A066897(n) - A066898(n) = A206563(n,1) - A206563(n,2). - Omar E. Pol, Mar 08 2012

G.f.: Sum_{j>0} x^j/(1+x^j)/Product_{k>0}(1 - x^k). - Emeric Deutsch, Feb 08 2016

a(n) = Sum_{i=1..n} (-1)^(i + 1)*A181187(n, i). - John M. Campbell, Mar 18 2018

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * Pi * sqrt(n)). - Vaclav Kotesovec, May 25 2018

For n > 0, a(n) = A305121(n) + A305123(n). - Vaclav Kotesovec, May 26 2018

a(n) = Sum_{k=-floor(n/2)+(n mod 2)..n} k * A240009(n,k). - Alois P. Heinz, Oct 23 2018

a(n) = Sum_{k>0} k * A264398(n,k). - Alois P. Heinz, Aug 05 2020

The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1], a total of 15 odd parts and 5 even parts, so that a(5)=10.

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

m:= irem(i, 2);

if n=0 then [1, 0, 0]

elif i<1 then [0, 0, 0]

else f:= b(n, i-1); g:= `if`(i>n, [0$3], b(n-i, i));

[f[1]+g[1], f[2]+g[2]+m*g[1], f[3]+g[3]+(1-m)*g[1]]

fi

end:

a:= n-> b(n, n)[2] -b(n, n)[3]:

seq(a(n), n=1..50); # Alois P. Heinz, Jul 09 2012

g := add(x^j/(1+x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 08 2016

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 *)

b[n_, i_] := b[n, i] = Module[{m, f, g}, m = Mod[i, 2]; If[n==0, {1, 0, 0}, If[i<1, {0, 0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + m*g[[1]], f[[3]] + g[[3]] + (1-m)* g[[1]]}]]]; a[n_] := b[n, n][[2]] - b[n, n][[3]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

Cf. A066897, A066898, A000041, A240009, A264398.

Sequence in context: A180964 A237668 A366754 * A357620 A185784 A185904

Adjacent sequences: A209420 A209421 A209422 * A209424 A209425 A209426

Clark Kimberling, Mar 08 2012

approved