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A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.

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%I A118712 #8 Feb 16 2025 08:33:01
%S A118712 1,0,0,-5,6,-16,9,-134400,647248,-1711908,6076067,-85248000,116477425,
%T A118712 -1764364437,909276004,-522319050599375232,14313181351994538493,
%U A118712 -165893335414907083200,2939566160282258664451,-5007637771411479278976,75399747694572065660672
%N A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.
%H A118712 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%e A118712 a(4) = -5 because of the determinant -5 =
%e A118712 |1,1,1,2|
%e A118712 |2,1,1,1|
%e A118712 |1,2,1,1|
%e A118712 |1,1,2,1|.
%e A118712 a(11) = 6076067 = determinant
%e A118712 |1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1|
%e A118712 |1, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2|
%e A118712 |2, 1, 1, 1, 1, 2, 1, 2, 1, 5, 2|
%e A118712 |2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 5|
%e A118712 |5, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1|
%e A118712 |1, 5, 2, 2, 1, 1, 1, 1, 2, 1, 2|
%e A118712 |2, 1, 5, 2, 2, 1, 1, 1, 1, 2, 1|
%e A118712 |1, 2, 1, 5, 2, 2, 1, 1, 1, 1, 2|
%e A118712 |2, 1, 2, 1, 5, 2, 2, 1, 1, 1, 1|
%e A118712 |1, 2, 1, 2, 1, 5, 2, 2, 1, 1, 1|
%e A118712 |1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 1|.
%o A118712 (GAP) A118712 := n -> DeterminantMat(List([0..n-1], i->List([0..n-1], j->NrSmallGroups(((j-i) mod n)+1)))); # _Eric M. Schmidt_, Nov 17 2013
%Y A118712 Cf. A000001, A048954, A052182, A066933, A086459, A086569.
%K A118712 sign
%O A118712 1,4
%A A118712 _Jonathan Vos Post_, May 20 2006
%E A118712 a(1) corrected by and more terms from _Eric M. Schmidt_, Nov 17 2013