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A284871
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Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols.
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8
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1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0
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OFFSET
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1,2
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COMMENTS
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A string and its reverse are considered to be equivalent.
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REFERENCES
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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]
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LINKS
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FORMULA
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T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.
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EXAMPLE
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Table starts:
1 2 3 4 5 6 7 8 ...
0 1 3 6 10 15 21 28 ...
0 4 15 36 70 120 189 280 ...
0 7 39 126 310 645 1197 2044 ...
0 18 132 540 1620 3990 8568 16632 ...
0 29 357 2034 7790 23295 58779 131012 ...
0 70 1131 8316 39370 140610 412965 1050616 ...
0 126 3276 32760 195300 839790 2882376 8388576 ...
...
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MATHEMATICA
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b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)
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PROG
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(PARI)
b(n, k) = (k^n + k^(ceil(n/2))) / 2;
a(n, k) = sumdiv(n, d, moebius(n/d) * b(d, k));
for(n=1, 10, for(k=1, 10, print1( a(n, k), ", "); ); print(); );
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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