A090794 Number of partitions of n such that the number of different parts is odd.

1, 2, 2, 3, 2, 5, 4, 9, 13, 19, 27, 43, 54, 71, 102, 124, 161, 200, 257, 319, 400, 484, 618, 761, 956, 1164, 1450, 1806, 2226, 2741, 3367, 4137, 5020, 6163, 7485, 9042, 10903, 13172, 15721, 18956, 22542, 26925, 31935, 37962, 44861, 53183, 62651

Offset: 1

Author

Vladeta Jovovic, Feb 12 2004

Keywords

• nonn
• easy

Examples

			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}, five of them have an odd number of elements, therefore a(6)=5.
		

Crossrefs

Cf. A060177, A002134, A000005, A027193, A090794.

Cf. also A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268

Cf. A092306.

Programs

Haskell

import Data.List (group)
a090794 = length . filter odd . 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

Formula

a(n) = b(n, 1, 0, 0) 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 off-diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [B(x), A(x); A(x), B(x)], where B(x) is the g.f. of A092306. - Peter Bala, Feb 10 2021

Extensions

More terms from Reinhard Zumkeller, Feb 17 2004

Definition simplified and shortened by Jonathan Sondow, Oct 13 2013