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A363451 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements.

%I A363451 #18 Oct 20 2023 08:52:02
%S A363451 1,0,2,2,9,23,99,353,1778,7927,45273,238797,1526331,9215950,65020448,
%T A363451 439742641,3388075807,25270974635,210763775071,1713657668021,
%U A363451 15359474721088,134902169999841,1291589459223627,12165062702520422,123780591852786693,1242763745129587332
%N A363451 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements.
%H A363451 Alois P. Heinz, <a href="/A363451/b363451.txt">Table of n, a(n) for n = 0..250</a>
%H A363451 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A363451 a(0) = 1: () the empty partition.
%e A363451 a(1) = 0.
%e A363451 a(2) = 2: 12, 1|2.
%e A363451 a(3) = 2: 123, 13|2.
%e A363451 a(4) = 9: 1234, 12|34, 12|3|4, 13|24, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
%e A363451 a(5) = 23: 12345, 123|45, 123|4|5, 125|34, 12|345, 125|3|4, 12|35|4, 134|25, 134|2|5, 135|24, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 15|23|4, 1|235|4, 145|2|3, 14|2|35, 15|2|34, 1|2|345, 15|2|3|4, 1|2|35|4.
%p A363451 b:= proc(n, x, y, m) option remember; `if`(n=0, `if`(x=y, 1, 0),
%p A363451       `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
%p A363451       `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
%p A363451     end:
%p A363451 a:= n-> b(n, 0$3):
%p A363451 seq(a(n), n=0..28);
%t A363451 b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, If[x == y, 1, 0], If[x + m > 0, b[n - 1, y, x, m]*(x + m), 0] + b[n - 1, y, x + 1, m] + If[y > 0, b[n - 1, y - 1, x, m + 1]*y, 0]];
%t A363451 a[n_] := b[n, 0, 0, 0];
%t A363451 Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Oct 20 2023, after _Alois P. Heinz_ *)
%Y A363451 Cf. A000110, A363434, A363435, A363454.
%K A363451 nonn
%O A363451 0,3
%A A363451 _Alois P. Heinz_, Jun 02 2023