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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. 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 Andrew Howroyd, Table of n, a(n) for n = 1..1275 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 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;   ... 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) 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 Columns 2-6 are A056278 (or A000740), A056279, A056280, A056281, A056282. Row sums are A082951. Cf. A054525, A008277, A327029. Sequence in context: A058150 A058151 A257673 * A248826 A058152 A058140 Adjacent sequences:  A137648 A137649 A137650 * A137652 A137653 A137654 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Feb 01 2008 EXTENSIONS Name changed and a(46)-a(66) from Andrew Howroyd, Aug 09 2018 STATUS approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)