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A363434 Total number of blocks containing only elements of the same parity in all partitions of [n].

%I A363434 #23 Sep 10 2023 09:47:38
%S A363434 0,1,2,7,24,97,412,1969,9898,54461,313944,1947613,12603100,86760255,
%T A363434 620559230,4682462777,36586620348,299664171115,2534306825064,
%U A363434 22355119509231,203115201624030,1917124624702475,18598998656476220,186822424157036439,1925326063016510832
%N A363434 Total number of blocks containing only elements of the same parity in all partitions of [n].
%H A363434 Alois P. Heinz, <a href="/A363434/b363434.txt">Table of n, a(n) for n = 0..250</a>
%H A363434 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363434 a(n) = Sum_{k=0..n} k * A124424(n,k).
%F A363434 a(n) = A363452(n) + A363453(n).
%F A363434 a(n) mod 2 = A000035(n).
%e A363434 a(3) = 7 = 0 + 1 + 2 + 1 + 3 : 123, 12|3, 13|2, 1|23, 1|2|3.
%p A363434 b:= proc(n, e, o, m) option remember; `if`(n=0, e+o,
%p A363434       (e+m)*b(n-1, o, e, m)+b(n-1, o, e+1, m)+
%p A363434        `if`(o=0, 0, o*b(n-1, o-1, e, m+1)))
%p A363434     end:
%p A363434 a:= n-> b(n, 0$3):
%p A363434 seq(a(n), n=0..24);
%t A363434 b[n_, e_, o_, m_] := b[n, e, o, m] = If[n == 0, e + o, (e + m)*b[n-1, o, e, m] + b[n - 1, o, e + 1, m] + If[o == 0, 0, o*b[n - 1, o - 1, e, m + 1]]];
%t A363434 a[n_] := b[n, 0, 0, 0];
%t A363434 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Sep 10 2023, after _Alois P. Heinz_ *)
%Y A363434 Cf. A000035, A124424, A363435, A363451, A363452, A363453.
%K A363434 nonn
%O A363434 0,3
%A A363434 _Alois P. Heinz_, Jun 01 2023