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A285037 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromic structures using exactly k different symbols, 1 <= k <= n/2 + 1.

%I A285037 #24 Nov 05 2019 13:52:55
%S A285037 1,0,1,0,1,0,2,1,0,3,1,0,4,5,1,0,7,6,1,0,10,18,7,1,0,14,25,10,1,0,21,
%T A285037 63,43,10,1,0,31,90,65,15,1,0,42,202,219,85,13,1,0,63,301,350,140,21,
%U A285037 1,0,91,650,1058,618,154,17,1,0,123,965,1701,1050,266,28,1
%N A285037 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromic structures using exactly k different symbols, 1 <= k <= n/2 + 1.
%C A285037 Permuting the symbols will not change the structure.
%C A285037 Equivalently, the number of n-bead aperiodic necklaces (Lyndon words) with exactly k symbols, up to permutation of the symbols, which when turned over are unchanged. When comparing with the turned over necklace a rotation is allowed but a permutation of the symbols is not.
%D A285037 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 A285037 Andrew Howroyd, <a href="/A285037/b285037.txt">Table of n, a(n) for n = 1..2600</a>
%F A285037 T(n, k) = Sum_{d | n} mu(n/d) * A285012(d, k).
%e A285037 Triangle starts:
%e A285037 1
%e A285037 0   1
%e A285037 0   1
%e A285037 0   2    1
%e A285037 0   3    1
%e A285037 0   4    5     1
%e A285037 0   7    6     1
%e A285037 0  10   18     7     1
%e A285037 0  14   25    10     1
%e A285037 0  21   63    43    10     1
%e A285037 0  31   90    65    15     1
%e A285037 0  42  202   219    85    13    1
%e A285037 0  63  301   350   140    21    1
%e A285037 0  91  650  1058   618   154   17   1
%e A285037 0 123  965  1701  1050   266   28   1
%e A285037 0 184 2016  4796  4064  1488  258  21  1
%e A285037 0 255 3025  7770  6951  2646  462  36  1
%e A285037 0 371 6220 21094 24914 12857 3222 410 26 1
%e A285037 0 511 9330 34105 42525 22827 5880 750 45 1
%e A285037 ...
%e A285037 Example for n=6, k=2:
%e A285037 There are 6 inequivalent solutions to A285012(6,2) which are 001100, 010010, 000100, 001010, 001110, 010101. Of these, 010010 and 010101 have a period less than 6, so T(6,2) = 6-2 = 4.
%o A285037 (PARI) \\ Ach is A304972
%o A285037 Ach(n,k=n) = {my(M=matrix(n, k, n, k, n>=k)); for(n=3, n, for(k=2, k, M[n, k]=k*M[n-2, k] + M[n-2, k-1] + if(k>2, M[n-2, k-2]))); M}
%o A285037 T(n,k=n\2+1) = {my(A=Ach(n\2+1,k), S=matrix(n\2+1, k, n, k, stirling(n,k,2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
%o A285037 { my(A=T(20)); for(n=1, matsize(A)[1], print(A[n,1..n\2+1])) } \\ _Andrew Howroyd_, Oct 01 2019
%o A285037 (PARI) \\ column sequence using above code.
%o A285037 ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ _Andrew Howroyd_, Oct 01 2019
%Y A285037 Columns 1..6 are: A063524, A056518, A056519, A056521, A056522, A056523.
%Y A285037 Partial row sums include A056513, A056514, A056515, A056516, A056517.
%Y A285037 Row sums are A285042.
%Y A285037 Cf. A284856, A284826, A284823, A285012, A304972.
%K A285037 nonn,tabf
%O A285037 1,7
%A A285037 _Andrew Howroyd_, Apr 08 2017