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A209423
Difference between the number of odd parts and the number of even parts in all the partitions of n.
12
1, 1, 4, 4, 10, 13, 24, 30, 52, 68, 105, 137, 202, 264, 376, 485, 669, 864, 1162, 1486, 1968, 2501, 3256, 4110, 5285, 6630, 8434, 10511, 13241, 16417, 20505, 25273, 31344, 38438, 47346, 57782, 70746, 85947, 104663, 126594, 153386, 184793, 222865, 267452
OFFSET
1,3
COMMENTS
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
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
FORMULA
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
EXAMPLE
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.
MAPLE
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
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 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 08 2012
STATUS
approved