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%I A369925 #7 Feb 06 2024 16:24:00
%S A369925 1,0,1,1,2,1,6,1,33,23,295,1,4877,1,44191,141210,749316,1,31762349,1,
%T A369925 309754506,3980911205,4704612121,1,1303743206944,55279816357,
%U A369925 2737023412201,343866841144704,564548508168226,1,145630899385513158,1,2359434158555273239
%N A369925 Number of uniform circular words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.
%C A369925 A word is uniform here if each symbol that occurs in the word occurs with the same frequency.
%C A369925 a(n) is the number of ways to partition [n] into parts of equal size and no part containing values that differ by 1 modulo n.
%H A369925 Andrew Howroyd, <a href="/A369925/b369925.txt">Table of n, a(n) for n = 0..200</a>
%F A369925 a(n) = Sum_{d|n} A369923(d, n/d} for n > 0.
%F A369925 a(p) = 1 for prime p.
%e A369925 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).
%e A369925 The a(6) = 6 words are ababab, abacbc, abcabc, abcacb, abcbac, abcdef.
%e A369925 The corresponding a(6) = 6 set partitions are:
%e A369925    {{1,3,5},{2,4,6}},
%e A369925    {{1,3},{2,5},{4,6}},
%e A369925    {{1,4},{2,5},{3,6}},
%e A369925    {{1,4},{2,6},{3,5}},
%e A369925    {{1,5},{2,4},{3,6}},
%e A369925    {{1},{2},{3},{4},{5},{6}}.
%o A369925 (PARI) \\ Needs T(n,k) from A369923.
%o A369925 a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}
%Y A369925 The case for adjacent elements possibly equal is A038041.
%Y A369925 Cf. A369923, A369924 (linear words).
%K A369925 nonn
%O A369925 0,5
%A A369925 _Andrew Howroyd_, Feb 06 2024

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