A092306 - OEIS

%I #31 Feb 11 2021 23:02:09

%S 1,0,0,1,2,5,6,11,13,17,23,29,34,47,64,74,107,136,185,233,308,392,518,

%T 637,814,1002,1272,1560,1912,2339,2863,3475,4212,5123,6147,7398,8935,

%U 10734,12843,15464,18382,22041,26249,31326,37213,44273,52375,62103,73376

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

%H Alois P. Heinz, <a href="/A092306/b092306.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F 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).

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

%F 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

%e 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.

%e 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.

%p b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,

%p        b(n, i-1, t) +add(b(n-i*j, i-1, 1-t), j=1..n/i)))

%p     end:

%p a:= n-> b(n, n, 1):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 29 2014

%t 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 *)

%o (Haskell)

%o import Data.List (group)

%o a092306 = length . filter even . map (length . group) . ps 1 where

%o    ps x 0 = [[]]

%o    ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]

%o -- _Reinhard Zumkeller_, Dec 19 2013

%Y Cf. A060177, A002133, A027187, A090794.

%K nonn,easy

%O 0,5

%A _Vladeta Jovovic_, Feb 12 2004

%E More terms from _Robert G. Wilson v_, Feb 16 2004