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A364267 Total number of blocks in all set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

%I A364267 #14 Jul 17 2023 09:14:29
%S A364267 0,1,2,5,9,24,55,169,454,1567,4823,18422,63609,265929,1014266,4595861,
%T A364267 19143089,93286964,420483103,2189786125,10601936382,58688597511,
%U A364267 303349005967,1776842374930,9754696729753,60223101819493,349624680839546,2267363687696309
%N A364267 Total number of blocks in all set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.
%C A364267 All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.  Blocks are ordered with increasing least elements.
%H A364267 Alois P. Heinz, <a href="/A364267/b364267.txt">Table of n, a(n) for n = 0..660</a>
%H A364267 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A364267 a(n) = Sum_{k=0..n} k * A274537(n,k).
%F A364267 a(n) mod 2 = A011655(n+1) for n>=1.
%e A364267 a(4) = 9 = 2 + 3 + 4: 13|24, 1|24|3, 1|2|3|4.
%e A364267 a(5) = 24 = 2 + 3 + 3 + 3 + 4 + 4 + 5: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
%p A364267 b:= proc(n, m, t) option remember; `if`(n=0, m, add(
%p A364267       `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
%p A364267     end:
%p A364267 a:= n-> b(n, 0, 1):
%p A364267 seq(a(n), n=0..30);
%Y A364267 Cf. A011655, A274537.
%K A364267 nonn
%O A364267 0,3
%A A364267 _Alois P. Heinz_, Jul 16 2023