Search: id:a137651
|
| |
|
|
A137651
|
|
Triangle read by rows: T(n,k) is the number of primitive (aperiodic) word structures of length n using exactly k different symbols.
|
|
+0
10
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
COMMENTS
|
Row sums = A082951: (1, 1, 4, 13, 51, 197, ...).
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = Sum{d|n} mu(n/d) * Stirling2(d, k). - Andrew Howroyd, Aug 09 2018
|
|
|
EXAMPLE
|
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;
...
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)
|
|
|
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 *)
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.004 seconds
|
