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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.
4
1, 1, 1, 2, 3, 7, 14, 39, 95, 304, 865, 3103, 10038, 39773, 143473, 620382, 2461099, 11504723, 49658054, 249102263, 1159930119, 6205900348, 30959905841, 175763987955, 934068692102, 5602484594053, 31563436487785, 199267671153562, 1185224170637619
OFFSET
0,4
COMMENTS
All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.
LINKS
FORMULA
a(n) = Sum_{k=0..n} A274537(n,k).
a(n) mod 2 = A011655(n) for n>=1.
EXAMPLE
a(3) = 2: 13|2, 1|2|3.
a(4) = 3: 13|24, 1|24|3, 1|2|3|4.
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.
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.
MAPLE
b:= proc(n, m, t) option remember; `if`(n=0, 1, add(
`if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);
MATHEMATICA
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}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 23 2018, translated from Maple *)
CROSSREFS
Row sums of A274537.
Column k=2 of A274835.
Cf. A011655.
Sequence in context: A296417 A296418 A006785 * A113182 A165433 A238728
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 27 2016
STATUS
approved