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A107424
Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.
12
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 17, 13, 3, 1, 0, 9, 43, 50, 20, 3, 1, 0, 16, 124, 220, 136, 36, 4, 1, 0, 28, 338, 866, 773, 296, 52, 4, 1, 0, 51, 941, 3435, 4280, 2303, 596, 78, 5, 1, 0, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 0, 170, 7234, 51061
OFFSET
1,8
COMMENTS
This classification is concerned with which beads are the same color, not with the colors themselves, so bbabcd is the same structure as aabacd. Cyclic permutations are also the same structure, e.g. abacda is also the same structure. However, order matters: the reverse of aabacd is equivalent to aabcad, which is also on the list.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
T(n, k) = Sum_{d|n} mu(n/d) * A152175(d, k). - Andrew Howroyd, Apr 09 2017
EXAMPLE
T(6, 4) = 13: {aaabcd, aabacd, aabcad, abacad, aabbcd, aabcbd, aabcdb, aacbbd, aacbdb, ababcd, abacbd, acabdb, abcabd}.
From Andrew Howroyd, Apr 09 2017 (Start)
Triangle starts:
1
0 1
0 1 1
0 2 2 1
0 3 5 2 1
0 5 17 13 3 1
0 9 43 50 20 3 1
0 16 124 220 136 36 4 1
0 28 338 866 773 296 52 4 1
0 51 941 3435 4280 2303 596 78 5 1
(End)
MATHEMATICA
A[d_, n_] := A[d, n] = Which[n == 0, 1, n == 1, DivisorSum[d, x^# &], d == 1, Sum[StirlingS2[n, k] x^k, {k, 0, n}], True, Expand[A[d, 1] A[d, n-1] + D[A[d, n-1], x] x]];
B[n_, k_] := Coefficient[DivisorSum[n, EulerPhi[#] A[#, n/#]&]/n/x, x, k];
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] B[#, k]&];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 06 2018, after Andrew Howroyd and Robert A. Russell *)
PROG
(PARI) \\ here R(n) is A152175 as square matrix.
R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n) = {my(M=R(n)); matrix(n, n, i, k, sumdiv(i, d, moebius(i/d)*M[d, k]))}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020
CROSSREFS
Columns 2-6 are A056303, A056304, A056305, A056306, A056307.
Partial row sums include A000048, A002075, A056300, A056301, A056302.
Row sums are A276547.
Sequence in context: A348967 A285308 A276543 * A155161 A185937 A292086
KEYWORD
nonn,tabl
AUTHOR
David Wasserman, May 26 2005
STATUS
approved