A092306 Number of partitions of n such that the set of parts has an even number of elements.

1, 0, 0, 1, 2, 5, 6, 11, 13, 17, 23, 29, 34, 47, 64, 74, 107, 136, 185, 233, 308, 392, 518, 637, 814, 1002, 1272, 1560, 1912, 2339, 2863, 3475, 4212, 5123, 6147, 7398, 8935, 10734, 12843, 15464, 18382, 22041, 26249, 31326, 37213, 44273, 52375, 62103, 73376

Offset: 0

Author

Vladeta Jovovic, Feb 12 2004

Keywords

• nonn
• easy

Examples

			The partitions of five are: {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}, The seven partitions have 1, 2, 2, 2, 2, 2 and 1 distinct parts respectively.
n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, six of them have an even number of elements, therefore a(6)=6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A060177, A002133, A027187, A090794.

Programs

Haskell

import Data.List (group)
a092306 = length . filter even . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       b(n, i-1, t) +add(b(n-i*j, i-1, 1-t), j=1..n/i)))
    end:
a:= n-> b(n, n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

Mathematica

f[n_] := Count[ Mod[ Length /@ Union /@ IntegerPartitions[n], 2], 0]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Feb 16 2004, updated by Jean-François Alcover, Jan 29 2014 *)

Formula

a(n) = b(n, 1, 0, 1) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004

G.f.: F(x)*G(x)/2, where F(x) = 1+Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).

a(n) = (A000041(n)+A104575(n))/2.

G.f. A(x) equals the main diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [A(x), B(x); B(x), A(x)], where B(x) is the g.f. of A090794. - Peter Bala, Feb 10 2021

Extensions

More terms from Robert G. Wilson v, Feb 16 2004