A275679 Number of set partitions of [n] with alternating block size parities.

1, 1, 1, 4, 3, 20, 43, 136, 711, 1606, 12653, 36852, 250673, 1212498, 6016715, 45081688, 196537387, 1789229594, 8963510621, 76863454428, 512264745473, 3744799424978, 32870550965259, 219159966518160, 2257073412153459, 15778075163815474, 165231652982941085

Offset: 0

Author

Alois P. Heinz, Aug 05 2016

Keywords

• nonn

Examples

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 3: 1234, 1|23|4, 1|24|3.
a(5) = 20: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 14|2|35, 15|2|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..592

Crossrefs

Cf. A003724, A005046, A007837, A038041, A275309, A275310, A275311, A275312, A275313, A286076, A361804.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      `if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
seq(a(n), n=0..35);

Mathematica

b[n_, t_] := b[n, t] = If[n==0, 1, Sum[If[OddQ[i+t], b[n-i, 1-t] * Binomial[n-1, i-1], 0], {i, 1, n}]]; a[n_] := If[n==0, 1, b[n, 0] + b[n, 1]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)