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A284826 Irregular triangle T(n,k) for 1 <= k <= (n+1)/2: T(n,k) = number of primitive (aperiodic) palindromic structures of length n using exactly k different symbols.

%I A284826 #28 Jun 12 2017 08:08:48
%S A284826 1,0,0,1,0,1,0,3,1,0,2,1,0,7,6,1,0,6,6,1,0,14,25,10,1,0,12,24,10,1,0,
%T A284826 31,90,65,15,1,0,27,89,65,15,1,0,63,301,350,140,21,1,0,56,295,349,140,
%U A284826 21,1,0,123,965,1701,1050,266,28,1
%N A284826 Irregular triangle T(n,k) for 1 <= k <= (n+1)/2: T(n,k) = number of primitive (aperiodic) palindromic structures of length n using exactly k different symbols.
%C A284826 Permuting the symbols will not change the structure.
%D A284826 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 A284826 Andrew Howroyd, <a href="/A284826/b284826.txt">Table of n, a(n) for n = 1..930</a>
%F A284826 T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284823(n, k-j)) / k!.
%F A284826 T(n, k) = Sum_{d | n} mu(n/d) * stirling2(ceiling(d/2), k).
%e A284826 Triangle starts:
%e A284826 1
%e A284826 0
%e A284826 0   1
%e A284826 0   1
%e A284826 0   3    1
%e A284826 0   2    1
%e A284826 0   7    6     1
%e A284826 0   6    6     1
%e A284826 0  14   25    10     1
%e A284826 0  12   24    10     1
%e A284826 0  31   90    65    15     1
%e A284826 0  27   89    65    15     1
%e A284826 0  63  301   350   140    21    1
%e A284826 0  56  295   349   140    21    1
%e A284826 0 123  965  1701  1050   266   28   1
%e A284826 0 120  960  1700  1050   266   28   1
%e A284826 0 255 3025  7770  6951  2646  462  36  1
%e A284826 0 238 2999  7760  6950  2646  462  36  1
%e A284826 0 511 9330 34105 42525 22827 5880 750 45 1
%e A284826 0 495 9305 34095 42524 22827 5880 750 45 1
%e A284826 --------------------------------------------
%e A284826 For n=5, structures with 2 symbols are aabaa, ababa and abbba, so T(5,2) = 3.
%e A284826 For n=6, structures with 2 symbols are aabbaa and abbbba, so T(6,2) = 2.
%e A284826 (In this case, the structure abaaba is excluded because it is not primitive.)
%t A284826 T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*StirlingS2[Ceiling[#/2], k]&];
%t A284826 Table[T[n, k], {n, 1, 15}, {k, 1, Floor[(n+1)/2]}] // Flatten (* _Jean-François Alcover_, Jun 12 2017, from 2nd formula *)
%o A284826 (PARI)
%o A284826 b(n,k) = sumdiv(n,d, moebius(n/d) * k^(ceil(d/2)));
%o A284826 a(n,k) = sum(j=0,k, b(n,k-j)*binomial(k,j)*(-1)^j)/k!;
%o A284826 for(n=1, 20, for(k=1, ceil(n/2), print1( a(n,k),", ");); print(););
%Y A284826 Columns 2-6 are A056481, A056482, A056483, A056484, A056485.
%Y A284826 Partial row sums include A056476, A056477, A056478, A056479, A056480.
%Y A284826 Row sums are A284841.
%Y A284826 Cf. A284823.
%K A284826 nonn,tabf
%O A284826 1,8
%A A284826 _Andrew Howroyd_, Apr 03 2017