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

%I A275679 #12 Jun 12 2023 18:53:44
%S A275679 1,1,1,4,3,20,43,136,711,1606,12653,36852,250673,1212498,6016715,
%T A275679 45081688,196537387,1789229594,8963510621,76863454428,512264745473,
%U A275679 3744799424978,32870550965259,219159966518160,2257073412153459,15778075163815474,165231652982941085
%N A275679 Number of set partitions of [n] with alternating block size parities.
%H A275679 Alois P. Heinz, <a href="/A275679/b275679.txt">Table of n, a(n) for n = 0..592</a>
%e A275679 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A275679 a(4) = 3: 1234, 1|23|4, 1|24|3.
%e A275679 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.
%p A275679 b:= proc(n, t) option remember; `if`(n=0, 1, add(
%p A275679       `if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
%p A275679     end:
%p A275679 a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
%p A275679 seq(a(n), n=0..35);
%t A275679 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 *)
%Y A275679 Cf. A003724, A005046, A007837, A038041, A275309, A275310, A275311, A275312, A275313, A286076, A361804.
%K A275679 nonn
%O A275679 0,4
%A A275679 _Alois P. Heinz_, Aug 05 2016