A137651 Triangle read by rows: T(n,k) is the number of primitive (aperiodic) word structures of length n using exactly k different symbols.
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, 1, 0, 120, 960, 1700, 1050, 266, 28, 1, 0, 252, 3024, 7770, 6951, 2646, 462, 36, 1, 0, 495, 9305, 34095, 42524, 22827, 5880, 750, 45, 1, 0, 1023, 28501, 145750, 246730, 179487, 63987, 11880, 1155, 55, 1
Offset: 1
Examples
First few rows of the triangle are:
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, 1;
0, 120, 960, 1700, 1050, 266, 28, 1;
...
From _Andrew Howroyd_, Apr 03 2017: (Start)
Primitive word structures are:
n=1: a => 1
n=2: ab => 1
n=3: aab, aba, abb; abc => 3 + 1
n=4: aaab, aaba, aabb, abaa, abba, abbb => 6 (k=2)
aabc, abac, abbc, abca, abcb, abcc => 6 (k=3)
(End)
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
Crossrefs
Programs
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Mathematica
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 *)
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PARI
T(n,k)={sumdiv(n, d, moebius(n/d)*stirling(d, k, 2))} \\ Andrew Howroyd, Aug 09 2018 -
Sage
# uses[DivisorTriangle from A327029] # Computes an additional column (1,0,0,...) # at the left hand side of the triangle. DivisorTriangle(moebius, stirling_number2, 10) # Peter Luschny, Aug 24 2019
Formula
A054525 * A008277 as infinite lower triangular matrices. A054525 = Mobius transform, A008277 = Stirling2 triangle.
T(n,k) = Sum{d|n} mu(n/d) * Stirling2(d, k). - Andrew Howroyd, Aug 09 2018
Extensions
Name changed and a(46)-a(66) from Andrew Howroyd, Aug 09 2018
Comments