OFFSET
1,8
COMMENTS
Permuting the symbols will not change the structure.
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..930
FORMULA
T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284823(n, k-j)) / k!.
T(n, k) = Sum_{d | n} mu(n/d) * stirling2(ceiling(d/2), k).
EXAMPLE
Triangle starts:
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 31 90 65 15 1
0 27 89 65 15 1
0 63 301 350 140 21 1
0 56 295 349 140 21 1
0 123 965 1701 1050 266 28 1
0 120 960 1700 1050 266 28 1
0 255 3025 7770 6951 2646 462 36 1
0 238 2999 7760 6950 2646 462 36 1
0 511 9330 34105 42525 22827 5880 750 45 1
0 495 9305 34095 42524 22827 5880 750 45 1
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For n=5, structures with 2 symbols are aabaa, ababa and abbba, so T(5,2) = 3.
For n=6, structures with 2 symbols are aabbaa and abbbba, so T(6,2) = 2.
(In this case, the structure abaaba is excluded because it is not primitive.)
MATHEMATICA
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*StirlingS2[Ceiling[#/2], k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, Floor[(n+1)/2]}] // Flatten (* Jean-François Alcover, Jun 12 2017, from 2nd formula *)
PROG
(PARI)
b(n, k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
a(n, k) = sum(j=0, k, b(n, k-j)*binomial(k, j)*(-1)^j)/k!;
for(n=1, 20, for(k=1, ceil(n/2), print1( a(n, k), ", "); ); print(); );
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Apr 03 2017
STATUS
approved