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A137651 Triangle read by rows: T(n,k) is the number of primitive (aperiodic) word structures of length n using exactly k different symbols.

%I A137651 #30 Mar 24 2020 12:37:29
%S A137651 1,0,1,0,3,1,0,6,6,1,0,15,25,10,1,0,27,89,65,15,1,0,63,301,350,140,21,
%T A137651 1,0,120,960,1700,1050,266,28,1,0,252,3024,7770,6951,2646,462,36,1,0,
%U A137651 495,9305,34095,42524,22827,5880,750,45,1,0,1023,28501,145750,246730,179487,63987,11880,1155,55,1
%N A137651 Triangle read by rows: T(n,k) is the number of primitive (aperiodic) word structures of length n using exactly k different symbols.
%C A137651 Row sums = A082951: (1, 1, 4, 13, 51, 197, ...).
%H A137651 Andrew Howroyd, <a href="/A137651/b137651.txt">Table of n, a(n) for n = 1..1275</a>
%F A137651 A054525 * A008277 as infinite lower triangular matrices. A054525 = Mobius transform, A008277 = Stirling2 triangle.
%F A137651 T(n,k) = Sum{d|n} mu(n/d) * Stirling2(d, k). - _Andrew Howroyd_, Aug 09 2018
%e A137651 First few rows of the triangle are:
%e A137651   1;
%e A137651   0,   1;
%e A137651   0,   3,   1;
%e A137651   0,   6,   6,    1;
%e A137651   0,  15,  25,   10,    1;
%e A137651   0,  27,  89,   65,   15,   1;
%e A137651   0,  63, 301,  350,  140,  21,  1;
%e A137651   0, 120, 960, 1700, 1050, 266, 28, 1;
%e A137651   ...
%e A137651 From _Andrew Howroyd_, Apr 03 2017: (Start)
%e A137651 Primitive word structures are:
%e A137651 n=1: a => 1
%e A137651 n=2: ab => 1
%e A137651 n=3: aab, aba, abb; abc => 3 + 1
%e A137651 n=4: aaab, aaba, aabb, abaa, abba, abbb => 6 (k=2)
%e A137651      aabc, abac, abbc, abca, abcb, abcc => 6 (k=3)
%e A137651 (End)
%t A137651 rows = 10; t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0]; A054525 = Array[t, {rows, rows}]; A008277 = Array[StirlingS2, {rows, rows}]; T = A054525 . A008277; Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 07 2017 *)
%o A137651 (PARI) T(n,k)={sumdiv(n, d, moebius(n/d)*stirling(d, k, 2))} \\ _Andrew Howroyd_, Aug 09 2018
%o A137651 (Sage) # uses[DivisorTriangle from A327029]
%o A137651 # Computes an additional column (1,0,0,...)
%o A137651 # at the left hand side of the triangle.
%o A137651 DivisorTriangle(moebius, stirling_number2, 10) # _Peter Luschny_, Aug 24 2019
%Y A137651 Columns 2-6 are A056278 (or A000740), A056279, A056280, A056281, A056282.
%Y A137651 Row sums are A082951.
%Y A137651 Cf. A054525, A008277, A327029.
%K A137651 nonn,tabl
%O A137651 1,5
%A A137651 _Gary W. Adamson_, Feb 01 2008
%E A137651 Name changed and a(46)-a(66) from _Andrew Howroyd_, Aug 09 2018