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

0,5

Links

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

Formula

a(n) = b(n, 1, 0, 1) with b(n, i, j, f) = if i<n then b(n-i, i, i, 1-f-(1-2*f)*0^(i-j)) + b(n, i+1, j, f) else (1-f-(1-2*f)*0^(i-j))*0^(i-n). - Reinhard 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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A060177, A002133, A027187, A090794.

Sequence in context: A337498 A045845 A002133 * A349158 A319242 A323398

Adjacent sequences: A092303 A092304 A092305 * A092307 A092308 A092309

Keyword

nonn,easy

Author

Vladeta Jovovic, Feb 12 2004

Extensions

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

Status

approved