|
%I #20 Oct 10 2019 22:20:00
%S 0,0,1,0,0,1,0,1,1,1,0,0,1,1,1,0,1,3,5,2,1,0,0,3,10,8,2,1,0,1,7,33,40,
%T 18,3,1,0,0,11,83,157,104,28,3,1,0,1,19,237,650,615,246,46,4,1,0,0,31,
%U 640,2522,3318,1857,495,65,4,1,0,1,63,1817,9888,17594,13311,4911,944,97,5,1
%N Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.
%C Permuting the colors does not change the necklace structure.
%C Equivalently, the number of k-block partitions of an n-set up to rotations where no block contains cyclically adjacent elements of the n-set.
%H Andrew Howroyd, <a href="/A327396/b327396.txt">Table of n, a(n) for n = 1..1275</a>
%e Triangle begins:
%e 0;
%e 0, 1;
%e 0, 0, 1;
%e 0, 1, 1, 1;
%e 0, 0, 1, 1, 1;
%e 0, 1, 3, 5, 2, 1;
%e 0, 0, 3, 10, 8, 2, 1;
%e 0, 1, 7, 33, 40, 18, 3, 1;
%e 0, 0, 11, 83, 157, 104, 28, 3, 1;
%e 0, 1, 19, 237, 650, 615, 246, 46, 4, 1;
%e 0, 0, 31, 640, 2522, 3318, 1857, 495, 65, 4, 1;
%e 0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1;
%e ...
%o (PARI)
%o R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace((y-1)*exp(-x + O(x*x^(n\m))) - y + exp(-x + sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d)) ), x, x^m))/x), -n)]))}
%o { my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Oct 09 2019
%Y Columns k=3..4 are A327397, A328130.
%Y Partial row sums include A306888, A309673.
%Y Row sums are A328150.
%Y Cf. A152175, A261139, A208535.
%K nonn,tabl
%O 1,18
%A _Andrew Howroyd_, Oct 04 2019
|
