A209797 The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2 and 1^22^1 in the pattern sense.

2, 6, 18, 56, 188, 695, 2838, 12726, 62140, 327760, 1854488, 11189273, 71627546, 484332314, 3446042310, 25712613664, 200599911596, 1632055365951, 13814906940846, 121414108567114, 1105838412755384, 10420517690466168, 101439025287805552, 1018689421191417393

Offset: 1

Author

Adam Goyt, Mar 13 2012

Keywords

• nonn

Comments

A partition of the set [n] is a family nonempty disjoint sets whose union is [n]. The blocks are written in order of increasing minima. A partition of the set [n] can be written as a word p=p_1p_2...p_n where p_i=j if element i is in block j. A partition q=q_1q_2...q_n contains partition p=p_1p_2...p_k if there is a subword q_{i_1}q_{i_2}...q_{i_k} such that q_{i_a}

Examples

			For n=2 the a(2)=6 solutions are 1^11^1, 1^21^1, 1^21^2, 1^12^1, 1^12^2, 1^22^2.
		

Formula

For n >=2, 2*B(n)+B(n-1)+sum(sum(B(n-j-k), k = 0 .. n-j), j = 2 .. n)+sum(B(j-1)*(B(n-j)+sum((k+binomial(n-j, k))*B(n-j-k), k = 1 .. n-j)), j = 2 .. n-1)

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