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a(n) is the number of trivial braids on 3 strands with 2*n crossings.
1

%I #21 Aug 03 2022 10:38:41

%S 1,4,28,244,2412,25804,290932,3403404,40914508

%N a(n) is the number of trivial braids on 3 strands with 2*n crossings.

%C In other words, a(n) is the number of products of 2*n generators in the braid group B_3 which are equal to the identity element of the group.

%C Only braids with an even number of crossings are considered because a braid with an odd number of crossings cannot be trivial.

%C If we do include the 0s corresponding to the odd values of the number of crossings, a group-theoretical name for this sequence is the cogrowth sequence of B_3.

%Y Cf. A000984 (number of trivial braids on 2 strands with 2*n crossings), A047849 (number of trivial permutations of 3 elements after 2*n adjacent transpositions).

%K nonn,more

%O 0,2

%A _Alexei Vernitski_, Jul 08 2022