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

1,2

Links

Table of n, a(n) for n=1..47.

Formula

a(n) = b(n, 1, 0, 0) 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 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

Example

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.

Prog

(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

Crossrefs

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

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

Cf. A092306.

Sequence in context: A113298 A058705 A218699 * A254858 A050323 A318286

Adjacent sequences: A090791 A090792 A090793 * A090795 A090796 A090797

Keyword

nonn,easy

Author

Vladeta Jovovic, Feb 12 2004

Extensions

More terms from Reinhard Zumkeller, Feb 17 2004

Definition simplified and shortened by Jonathan Sondow, Oct 13 2013

Status

approved