%I #28 Jun 12 2017 08:08:48

%S 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 31,90,65,15,1,0,27,89,65,15,1,0,63,301,350,140,21,1,0,56,295,349,140,

%U 21,1,0,123,965,1701,1050,266,28,1

%N 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 Permuting the symbols will not change the structure.

%D 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 Andrew Howroyd, <a href="/A284826/b284826.txt">Table of n, a(n) for n = 1..930</a>

%F T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284823(n, k-j)) / k!.

%F T(n, k) = Sum_{d | n} mu(n/d) * stirling2(ceiling(d/2), k).

%e Triangle starts:

%e 1

%e 0

%e 0 1

%e 0 1

%e 0 3 1

%e 0 2 1

%e 0 7 6 1

%e 0 6 6 1

%e 0 14 25 10 1

%e 0 12 24 10 1

%e 0 31 90 65 15 1

%e 0 27 89 65 15 1

%e 0 63 301 350 140 21 1

%e 0 56 295 349 140 21 1

%e 0 123 965 1701 1050 266 28 1

%e 0 120 960 1700 1050 266 28 1

%e 0 255 3025 7770 6951 2646 462 36 1

%e 0 238 2999 7760 6950 2646 462 36 1

%e 0 511 9330 34105 42525 22827 5880 750 45 1

%e 0 495 9305 34095 42524 22827 5880 750 45 1

%e --------------------------------------------

%e For n=5, structures with 2 symbols are aabaa, ababa and abbba, so T(5,2) = 3.

%e For n=6, structures with 2 symbols are aabbaa and abbbba, so T(6,2) = 2.

%e (In this case, the structure abaaba is excluded because it is not primitive.)

%t T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*StirlingS2[Ceiling[#/2], k]&];

%t Table[T[n, k], {n, 1, 15}, {k, 1, Floor[(n+1)/2]}] // Flatten (* _Jean-François Alcover_, Jun 12 2017, from 2nd formula *)

%o (PARI)

%o b(n,k) = sumdiv(n,d, moebius(n/d) * k^(ceil(d/2)));

%o a(n,k) = sum(j=0,k, b(n,k-j)*binomial(k,j)*(-1)^j)/k!;

%o for(n=1, 20, for(k=1, ceil(n/2), print1( a(n,k),", ");); print(););

%Y Columns 2-6 are A056481, A056482, A056483, A056484, A056485.

%Y Partial row sums include A056476, A056477, A056478, A056479, A056480.

%Y Row sums are A284841.

%Y Cf. A284823.

%K nonn,tabf

%O 1,8

%A _Andrew Howroyd_, Apr 03 2017