A276544 Triangle read by rows: T(n,k) = number of primitive (aperiodic) reversible string structures with n beads using exactly k different colors.
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
Examples
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)
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)
Crossrefs
Programs
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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 *) -
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
Formula
T(n, k) = Sum_{d|n} mu(n/d) * A284949(d, k).
Comments