A369925 Number of uniform circular words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.
1, 0, 1, 1, 2, 1, 6, 1, 33, 23, 295, 1, 4877, 1, 44191, 141210, 749316, 1, 31762349, 1, 309754506, 3980911205, 4704612121, 1, 1303743206944, 55279816357, 2737023412201, 343866841144704, 564548508168226, 1, 145630899385513158, 1, 2359434158555273239
Offset: 0
Keywords
Examples
a(1) = 0 because the symbol 'a' is considered to be adjacent to itself in a circular word. The set partition {{1}} is also excluded because 1 == 1 + 1 (mod 1).
The a(6) = 6 words are ababab, abacbc, abcabc, abcacb, abcbac, abcdef.
The corresponding a(6) = 6 set partitions are:
{{1,3,5},{2,4,6}},
{{1,3},{2,5},{4,6}},
{{1,4},{2,5},{3,6}},
{{1,4},{2,6},{3,5}},
{{1,5},{2,4},{3,6}},
{{1},{2},{3},{4},{5},{6}}.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
Crossrefs
Programs
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PARI
\\ Needs T(n,k) from A369923. a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}
Formula
a(n) = Sum_{d|n} A369923(d, n/d) for n > 0.
a(p) = 1 for prime p.
Comments