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A369079 Number of partitions of [n] such that the element sum of each block is odd.

%I A369079 #22 Sep 02 2024 21:54:34
%S A369079 1,1,1,2,4,10,28,96,320,1436,5556,28768,129600,730864,3756936,
%T A369079 23286784,132872192,910013776,5679982288,42235062784,286769980416,
%U A369079 2281079563104,16732506817280,141975748567040,1115928688967680,10077454948692288,84383735744758464
%N A369079 Number of partitions of [n] such that the element sum of each block is odd.
%C A369079 Number of partitions of [n] such that each block has an odd number of odd elements.
%H A369079 Alois P. Heinz, <a href="/A369079/b369079.txt">Table of n, a(n) for n = 0..592</a>
%H A369079 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A369079 a(0) = 1: the empty partition.
%e A369079 a(1) = 1: 1.
%e A369079 a(2) = 1: 12.
%e A369079 a(3) = 2: 12|3, 1|23.
%e A369079 a(4) = 4: 124|3, 12|34, 14|23, 1|234.
%e A369079 a(5) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
%p A369079 b:= proc(n, x, y) option remember; `if`(n=0, `if`(y=0, 1, 0),
%p A369079      `if`(n::odd, b(n-1, x+1, y)+`if`(x>0, x*b(n-1, x-1, y+1), 0)+
%p A369079      `if`(y>0, y*b(n-1, x+1, y-1), 0), b(n-1, x, y+1)+(x+y)*b(n-1, x, y)))
%p A369079     end:
%p A369079 a:= n-> b(n, 0$2):
%p A369079 seq(a(n), n=0..26);
%p A369079 # second Maple program:
%p A369079 b:= proc(x, y) option remember; `if`(x+y=0, 1,
%p A369079       add(`if`(j::odd, binomial(x-1, j-1)*add(
%p A369079       b(x-j, y-i)*binomial(y, i), i=0..y), 0), j=1..x))
%p A369079     end:
%p A369079 a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
%p A369079 seq(a(n), n=0..26);
%Y A369079 Column k=2 of A375924.
%Y A369079 Cf. A000009, A000110, A152140, A290383, A369080, A374692.
%K A369079 nonn
%O A369079 0,4
%A A369079 _Alois P. Heinz_, Jan 12 2024