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A347546 Number of involutions of doubly alternating Baxter permutations of length n.

%I A347546 #20 Jun 29 2022 02:50:48
%S A347546 1,1,1,1,2,2,3,5,8,12,16,32,44,84,105,231,292,636,768,1792,2166,5080,
%T A347546 6012,14592,17234,42198,49336,123088,143536,361190,418971,1066497,
%U A347546 1234242,3164870,3651296,9436968,10866726,28255468,32469716,84925632,97443786,256131058
%N A347546 Number of involutions of doubly alternating Baxter permutations of length n.
%H A347546 Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, <a href="https://arxiv.org/abs/2206.13877">Pattern avoiding alternating involutions</a>, arXiv:2206.13877 [math.CO], 2022.
%H A347546 Sook Min, <a href="https://doi.org/10.14403/jcms.2021.34.3.253">The Enumeration of Involutions of Doubly Alternating Baxter Permutations</a>, Journal of the Chungcheong Mathematical Society, 34(3) (2021), 253-257.
%o A347546 (Python)
%o A347546 def b(n):
%o A347546     if (0<=n<=3):
%o A347546         return 1
%o A347546     if (n==4):
%o A347546         return 2
%o A347546     if (n%2==1):
%o A347546       t=0
%o A347546       for k in range(1, ((n+1)//2)):
%o A347546           t+=b(2*k-2)*b(n-2*k)
%o A347546       return t
%o A347546     else:
%o A347546         s=0
%o A347546         for j in range(round(n/4), (n//2)):
%o A347546             s+=b(4*j-n)*b(n-2*j-1)
%o A347546         return b(n-1)+s
%o A347546 for i in range(30):
%o A347546     print(str(i)+': '+str(b(i)))
%Y A347546 Cf. A001181.
%K A347546 nonn
%O A347546 0,5
%A A347546 _Sook Min_, Sep 06 2021