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A274538 Number of set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

%I A274538 #16 Jul 16 2023 16:20:33
%S A274538 1,1,1,2,3,7,14,39,95,304,865,3103,10038,39773,143473,620382,2461099,
%T A274538 11504723,49658054,249102263,1159930119,6205900348,30959905841,
%U A274538 175763987955,934068692102,5602484594053,31563436487785,199267671153562,1185224170637619
%N A274538 Number of set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.
%C A274538 All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.
%H A274538 Alois P. Heinz, <a href="/A274538/b274538.txt">Table of n, a(n) for n = 0..662</a>
%H A274538 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A274538 a(n) = Sum_{k=0..n} A274537(n,k).
%F A274538 a(n) mod 2 = A011655(n) for n>=1.
%e A274538 a(3) = 2: 13|2, 1|2|3.
%e A274538 a(4) = 3: 13|24, 1|24|3, 1|2|3|4.
%e A274538 a(5) = 7: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
%e A274538 a(6) = 14: 135|246, 13|246|5, 13|24|5|6, 15|246|3, 15|24|3|6, 1|246|35, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|26|3|4|5, 1|2|3|46|5, 1|2|3|4|5|6.
%p A274538 b:= proc(n, m, t) option remember; `if`(n=0, 1, add(
%p A274538       `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
%p A274538     end:
%p A274538 a:= n-> b(n, 0, 1):
%p A274538 seq(a(n), n=0..30);
%t A274538 b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j, 2] == t, b[n - 1, Max[m, j], 1 - t], 0], {j, 1, m + 1}]];
%t A274538 a[n_] := b[n, 0, 1];
%t A274538 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 23 2018, translated from Maple *)
%Y A274538 Row sums of A274537.
%Y A274538 Column k=2 of A274835.
%Y A274538 Cf. A011655.
%K A274538 nonn
%O A274538 0,4
%A A274538 _Alois P. Heinz_, Jun 27 2016