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A276544
Triangle read by rows: T(n,k) = number of primitive (aperiodic) reversible string structures with n beads using exactly k different colors.
10
1, 0, 1, 0, 2, 1, 0, 4, 4, 1, 0, 9, 15, 6, 1, 0, 16, 49, 37, 9, 1, 0, 35, 160, 183, 76, 12, 1, 0, 66, 498, 876, 542, 142, 16, 1, 0, 133, 1544, 3930, 3523, 1346, 242, 20, 1, 0, 261, 4715, 17179, 21392, 11511, 2980, 390, 25, 1
OFFSET
1,5
COMMENTS
A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
REFERENCES
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
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) * A284949(d, k).
EXAMPLE
Triangle starts
1
0 1
0 2 1
0 4 4 1
0 9 15 6 1
0 16 49 37 9 1
0 35 160 183 76 12 1
0 66 498 876 542 142 16 1
0 133 1544 3930 3523 1346 242 20 1
0 261 4715 17179 21392 11511 2980 390 25 1
...
Primitive reversible word structures are:
n=1: a => 1
n=2: ab => 1
n=3: aab, aba; abc => 2 + 1
n=4: aaab, aaba, aabb, abba => 4 (k=2)
aabc, abac, abbc, abca => 4 (k=3)
MATHEMATICA
Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[n == 0, 1, 0], 1, If[n > 0, 1, 0], _, If[OddQ[n], Sum[Binomial[(n - 1)/2, i] Ach[n - 1 - 2 i, k - 1], {i, 0, (n - 1)/2}], Sum[Binomial[n/2 - 1, i] (Ach[n - 2 - 2 i, k - 1] + 2^i Ach[n - 2 - 2 i, k - 2]), {i, 0, n/2 - 1}]]]
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] (StirlingS2[#, k] + Ach[#, k])/2& ];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 29 2018, after Robert A. Russell and Andrew Howroyd *)
PROG
(PARI) \\ here Ach is A304972 as matrix.
Ach(n, m=n)={my(M=matrix(n, m, i, k, i>=k)); for(i=3, n, for(k=2, m, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
T(n, m=n)={my(M=matrix(n, m, i, k, stirling(i, k, 2)) + Ach(n, m)); matrix(n, m, i, k, sumdiv(i, d, moebius(i/d)*M[d, k]))/2}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020
CROSSREFS
Columns 2-6 are A056336, A056337, A056338, A056339, A056340.
Partial row sums include A056331, A056332, A056333, A056334, A056335.
Row sums are A276549.
Sequence in context: A244124 A183190 A296129 * A214753 A158454 A049243
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 09 2017
STATUS
approved