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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.

%I A276544 #20 Jan 10 2020 05:31:34
%S A276544 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,
%T A276544 66,498,876,542,142,16,1,0,133,1544,3930,3523,1346,242,20,1,0,261,
%U A276544 4715,17179,21392,11511,2980,390,25,1
%N A276544 Triangle read by rows: T(n,k) = number of primitive (aperiodic) reversible string structures with n beads using exactly k different colors.
%C A276544 A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
%D A276544 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]
%H A276544 Andrew Howroyd, <a href="/A276544/b276544.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A276544 T(n, k) = Sum_{d|n} mu(n/d) * A284949(d, k).
%e A276544 Triangle starts
%e A276544 1
%e A276544 0   1
%e A276544 0   2    1
%e A276544 0   4    4     1
%e A276544 0   9   15     6     1
%e A276544 0  16   49    37     9     1
%e A276544 0  35  160   183    76    12    1
%e A276544 0  66  498   876   542   142   16   1
%e A276544 0 133 1544  3930  3523  1346  242  20  1
%e A276544 0 261 4715 17179 21392 11511 2980 390 25 1
%e A276544 ...
%e A276544 Primitive reversible word structures are:
%e A276544 n=1: a => 1
%e A276544 n=2: ab => 1
%e A276544 n=3: aab, aba; abc => 2 + 1
%e A276544 n=4: aaab, aaba, aabb, abba => 4 (k=2)
%e A276544      aabc, abac, abbc, abca => 4 (k=3)
%t A276544 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 A276544 T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] (StirlingS2[#, k] + Ach[#, k])/2& ];
%t A276544 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 29 2018, after _Robert A. Russell_ and _Andrew Howroyd_ *)
%o A276544 (PARI) \\ here Ach is A304972 as matrix.
%o A276544 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}
%o A276544 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}
%o A276544 { my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Jan 09 2020
%Y A276544 Columns 2-6 are A056336, A056337, A056338, A056339, A056340.
%Y A276544 Partial row sums include A056331, A056332, A056333, A056334, A056335.
%Y A276544 Row sums are A276549.
%Y A276544 Cf. A284871, A284949, A277504, A137651, A107424, A276543.
%K A276544 nonn,tabl
%O A276544 1,5
%A A276544 _Andrew Howroyd_, Apr 09 2017