A011795 -id:A011795 - cp's OEIS frontend

A051168 Triangular array T(h,k) for 0 <= k <= h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 1, 3, 7, 8, 7, 3, 1, 0, 0, 1, 4, 9, 14, 14, 9, 4, 1, 0, 0, 1, 4, 12, 20, 25, 20, 12, 4, 1, 0, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 0, 0, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 0, 0, 1, 6

Offset: 0

Clark Kimberling

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.
T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.
From R. J. Mathar, Jul 31 2008: (Start)
This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:
This array begins as follows:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63
0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330
0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463
0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598
0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228
0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060
0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175
0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248
0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885
0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842
0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220
0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440
...
It is essentially symmetric: A(r,r+i) = A(r,r-i+1).
Some of the diagonals are:
A(r,r+1): A000108
A(r,r): A022553
A(r,r-1): A000108
A(r,r+2): A000150
A(r,r+3): A050181
A(r,r+4): A050182
A(r,r+5): A050183
A(r,r-2): A000150 (End)
Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019

			Triangle begins with:
h=0: 1
h=1: 1, 1
h=2: 0, 1, 0
h=3: 0, 1, 1, 0
h=4: 0, 1, 1, 1,  0
h=5: 0, 1, 2, 2,  1,  0
h=6: 0, 1, 2, 3,  2,  1, 0
h=7: 0, 1, 3, 5,  5,  3, 1, 0
h=8: 0, 1, 3, 7,  8,  7, 3, 1, 0
h=9: 0, 1, 4, 9, 14, 14, 9, 4, 1, 0
...
T(6,3) counts classes {111000},{110100},{110010}, each of 6 aperiodic. The class {100100} contains 3 periodic words, counted by T(3,1) as {100}, consisting of 3 aperiodic words 100,010,001.

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238; MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188; MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
P. Stanica and S. Maitra, Rotation symmetric boolean functions - count and cryptographic properties, Disc. Appl. Math. 156 (2008) 1567-1580; see h_{n,w} in eq. (3).
Index entries for sequences related to Lyndon words

Columns 1-11: A000012, A004526(n-1), A001840(n-4), A006918(n-4), A011795(n-1), A011796(n-6), A011797(n-1), A031164(n-9), A011845, A032168, A032169. See also A000150.

Cf. A047996, A052307, A052314, A092964, A185158, A123223, A124814.

A := proc(r,n) local gf,d,genf; genf := 1/(1-x) ; gf := 0 ; for d in numtheory[divisors](r) do gf := gf + numtheory[mobius](d)*(subs(x= x^d,genf))^(r/d) ; od: gf := expand(gf/r) ; coeftayl(gf,x=0,n) ; end proc:
A051168 := proc(n,k) if n<=1 then 1; elif n=0 or n=k then 0; else A(n-k,k) ; end if;
end proc:
seq(seq(A051168(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 29 2011

Table[If[n===0,1,1/n Plus@@(MoebiusMu[ # ]Binomial[n/#,k/# ]&/@ Divisors[GCD[n,k]])],{n,0,12},{k,0,n}] (* Wouter Meeussen, Jul 20 2008 *)

{T(n, k) = local(A, ps, c); if( k<0 || k>n, 0, if( n==0 && k==0, 1, A = x * O(x^n) + y * O(y^n); ps = 1 - x - y + A; for( m=1, n, for( i=0, m, c = polcoeff( polcoeff(ps, i, x), m-i, y); if( m==n && i==k, break(2), ps *= (1 -y^(m-i) * x^i + A)^c))); -c))} /* Michael Somos, Jul 03 2004 */

T(n,k) = if (n==0, 1, (1/n) * sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d,k/d)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 16 2018

T(h, k) = 1 for (h, k) in {(0, 0), (1, 0), (1, 1)}; T(h, k) = 0 if h >= 2 and k = 0 or k = h. Otherwise, T(h, k) = (1/h)*(C(h, k)-S(h, k)), where S(h, k) = Sum_{d <= 2, d|h, d|k} (h/d)*T(h/d, k/d).
1 - x - y = Product_{i,j} (1 - x^i * y^j)^T(i+j, j) where i >= 0, j >= 0 are not both zero. - Michael Somos, Jul 03 2004
The prime rows are given by (1+x)^p/p with noninteger coefficients rounded to zero. E.g., for h = 2 below, (1 + x)^2/2 = (1 + 2*x + x^2)/2 = 0.5 + x + 0.5*x^2 gives (0,1,0). - Tom Copeland, Oct 21 2014
T(n,k) = (1/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d, k/d), for n > 0. - Andrew Howroyd, Mar 26 2017
From Petros Hadjicostas, Jun 16 2019: (Start)
O.g.f. for column k >= 1: (x^k/k) * Sum_{d|k} mu(d)/(1 - x^d)^(k/d).
Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = 1 - Sum_{d >= 1} (mu(d)/d) *log(1 - x^d * (1 + y^d)).
(End)

A011847 Triangle of numbers read by rows: T(n,k) = floor( C(n,k)/(k+1) ), where k=0..n-1 and n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 21, 25, 21, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 41, 66, 77, 66, 41, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1, 6, 26, 71, 143, 214, 245, 214, 143, 71, 26, 6, 1

Offset: 1

N. J. A. Sloane

When k+1 is a prime >= 2, then T(n,k) = floor(C(n,k)/(k+1)) is the number of aperiodic necklaces of n+1 beads of 2 colors such that k+1 of them are black and n-k of them are white. This is not true when k+1 is a composite >= 4. For more details, see the comments for sequences A032168 and A032169. - Petros Hadjicostas, Aug 27 2018

Differs from A245558 from row n = 9, k = 4 on. - M. F. Hasler, Sep 29 2018

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   1;
  1, 2,  3,   2,   1;
  1, 3,  5,   5,   3,   1;
  1, 3,  7,   8,   7,   3,    1;
  1, 4,  9,  14,  14,   9,    4,    1;
  1, 4, 12,  21,  25,  21,   12,    4,    1;
  1, 5, 15,  30,  42,  42,   30,   15,    5,    1;
  1, 5, 18,  41,  66,  77,   66,   41,   18,    5,   1;
  1, 6, 22,  55,  99, 132,  132,   99,   55,   22,   6,   1;
  1, 6, 26,  71, 143, 214,  245,  214,  143,   71,  26,   6,   1;
  1, 7, 30,  91, 200, 333,  429,  429,  333,  200,  91,  30,   7,  1;
  1, 7, 35, 113, 273, 500,  715,  804,  715,  500, 273, 113,  35,  7, 1;
  1, 8, 40, 140, 364, 728, 1144, 1430, 1430, 1144, 728, 364, 140, 40, 8, 1;
...
More than the usual number of rows are shown in order to distinguish this triangle from A245558, from which it differs in rows 9, 11, 13, ....
From _Petros Hadjicostas_, Aug 27 2018: (Start)
For k+1 = 2 and n >= k+1 = 2, the n-th element of column k=1 above, [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] (i.e., the number A008619(n-2) = floor(n/2)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 2 of them are black and n-1 of them are white. (The offset of sequence A008619 is 0.)
For k+1 = 3 and n >= k+1 = 3, the n-th element of column k=2 above, [0, 0, 1, 2, 3, 5, 7, 9, 12, ...] (i.e., the number A001840(n-2) = floor(C(n,2)/3)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 3 of them are black and n-2 of them are white. (The offset of sequence A001840 is 0.)
For k+1 = 5 and n >= k+1 = 5, the n-th element of column k=4 above, [0, 0, 0, 0, 1, 3, 7, 14, 25, 42, ... ] (i.e., the number A011795(n) = floor(C(n,4)/5)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 5 of them are black and n-4 of them are white. (The offset of sequence A011795 is 0.)
Counterexample for k+1 = 4: It can be proved that, for n >= k+1 = 4, the number of aperiodic necklaces of n+1 beads of 2 colors such that 4 of them are black and n-3 of them are white is (1/4)*Sum_{d|4} mu(d)*I(d|n+1)*C(floor((n+1)/d) - 1, 4/d - 1) = (1/4)*(C(n, 3) - I(2|n+1)*floor((n-1)/2)), where I(a|b) = 1 if integer a divides integer b, and 0 otherwise. For n odd >= 9, the number (1/4)*(C(n, 3) - I(2|n+1)*floor((n-1)/2)) = A006918(n-3) is not equal to floor(C(n,3)/4) = A011842(n).
(End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Columns include A008619, A001840, A011842, A011795, A011843, A011797, A011844, A011845, A011846, A032169.
Sums: A095718 (row), A095719 (diagonal).
Cf. A006918, A245558.

A011847:= func< n,k | Floor(Binomial(n+1,k+1)/(n+1)) >;
[A011847(n,k): k in [0..n-1], n in [1..20]]; // G. C. Greubel, Oct 20 2024

Table[Floor[Binomial[n,k]/(k+1)],{n,20},{k,0,n-1}]//Flatten (* Harvey P. Dale, Jan 09 2019 *)

A011847(n,k)=binomial(n,k)\(k+1) \\ M. F. Hasler, Sep 30 2018

def A011847(n,k): return binomial(n+1,k+1)//(n+1)
flatten([[A011847(n,k) for k in range(n)] for n in range(1,21)]) # G. C. Greubel, Oct 20 2024

The rows are palindromic: T(n, k) = T(n, n-k-1) for n >= 1 and 0 <= k <= n-1.

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A095719(n). - G. C. Greubel, Oct 20 2024

A011915 a(n) = floor(n(n-1)(n-2)*(n-3)/5).

Original entry on oeis.org

0, 0, 0, 0, 4, 24, 72, 168, 336, 604, 1008, 1584, 2376, 3432, 4804, 6552, 8736, 11424, 14688, 18604, 23256, 28728, 35112, 42504, 51004, 60720, 71760, 84240, 98280, 114004, 131544, 151032, 172608, 196416, 222604, 251328, 282744, 317016, 354312

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Sequences of the form floor(24*binomial(n,4)/m): A052762 (m=1), A033486 (m=2), A162668 (m=3), A033487 (m=4), this sequence (m=5), A033488 (m=6), A011917 (m=7), A050534 (m=8), A011919 (m=9), 2*A011930 (m=10), A011921 (m=11), A034827 (m=12), A011923 (m=13), A011924 (m=14), A011925 (m=15), A011926 (m=16), A011927 (m=17), A011928 (m=18), A011929 (m=19), A011930 (m=20), A011931 (m=21), A011932 (m=22), A011933 (m=23), A000332 (m=24), A011935 (m=25),A011936 (m=26), A011937 (m=27), A011938 (m=28), A011939 (m=29), A011940 (m=30), A011941 (m=31), A011942 (m=32), A011795 (m=120).

[Floor(n*(n-1)*(n-2)*(n-3)/5): n in [0..60]]; // Vincenzo Librandi, Jun 19 2012

Table[Floor[n(n-1)(n-2)(n-3)/5], {n,60}] (* Stefan Steinerberger, Apr 10 2006 *)
CoefficientList[Series[4*x^4*(1+2*x+2*x^3+x^4)/((1-x)^4*(1+x^5)),{x,0,60}],x] (* Vincenzo Librandi, Jun 19 2012 *)

[24*binomial(n,4)//5 for n in range(61)] # G. C. Greubel, Oct 20 2024

a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +a(n-5) -4*a(n-6) +6*a(n-7) -4*a(n-8) +a(n-9).
G.f.: 4*x^4*(1+2*x+2*x^3+x^4) / ( (1-x)^5*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Apr 15 2010
a(n) = 4*A011930(n). - G. C. Greubel, Oct 20 2024

More terms from Stefan Steinerberger, Apr 10 2006

Zero added in front by R. J. Mathar, Apr 15 2010

A124813 Number of 4-ary Lyndon words of length n with exactly five 1s.

Original entry on oeis.org

3, 27, 189, 1134, 6123, 30618, 144342, 649539, 2814669, 11821608, 48361131, 193444524, 758897748, 2927177028, 11123272701, 41712272649, 154580775111, 566796175407, 2058365058057, 7410114208989, 26464693603590, 93829368230910

Offset: 6

Mike Zabrocki, Nov 08 2006

			a(7) = 27 because 11111ab, 1111a1b, 111a11b for a,b=2,3,4 are all Lyndon of length 7.

G. C. Greubel, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,246,-45,270,-810,1215,-729).

Cf. A011795, A124723, A124810, A124811, A124812, A124814.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 3*x^6*(1-6*x+18*x^2-27*x^3+16*x^4)/((1-3*x)^5*(1-3*x^5)) )); // G. C. Greubel, Aug 17 2023

3*(1 -6*x +18*x^2 -27*x^3 +16*x^4)/((1-3*x)^5*(1-3*x^5)) + O[x]^22 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2017 *)

def f(x): return 3*x^6*(1-6*x+18*x^2-27*x^3+16*x^4)/((1-3*x)^5*(1-3*x^5))
def A124813_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
a=A124813_list(46); a[6:] # G. C. Greubel, Aug 17 2023

O.g.f.: 3*x^6*(1 - 6*x + 18*x^2 - 27*x^3 + 16*x^4)/((1 - 3*x)^5*(1 - 3*x^5)).
O.g.f.: (1/5)*((x/(1-3*x))^5 - x^5/(1-3*x^5)).
a(n) = (1/5)*Sum_{d|5, d|n} mu(d)*C(n/d-1, (n-5)/d )*3^((n-5)/d).
a(n) = (1/5)*C(n-1, 4)*3^(n-5) if n=1,2,3,4 mod 5.
a(n) = (1/5)*C(n-1, 4)*3^(n-5) - (1/5)*3^((n-5)/5) if n=0 mod 5.

A051170 T(n,5), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 5 black beads and n-5 white beads.

Original entry on oeis.org

0, 1, 3, 7, 14, 25, 42, 66, 99, 143, 200, 273, 364, 476, 612, 775, 969, 1197, 1463, 1771, 2125, 2530, 2990, 3510, 4095, 4750, 5481, 6293, 7192, 8184, 9275, 10472, 11781, 13209, 14763, 16450, 18278, 20254, 22386, 24682

Offset: 5

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 5..5000
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A000031, A001037, A051168. Same as A011795(n-1).

First differences of A036837.

[ Floor(Binomial(n,5)/n): n in [5..30]]; // G. C. Greubel, Nov 26 2017

Table[Floor[Binomial[n, 5]/n], {n, 5, 50}] (* G. C. Greubel, Nov 26 2017 *)

for(n=5,30, print1(floor(binomial(n,5)/n), ", ")) \\ G. C. Greubel, Nov 26 2017

G.f.: -x^6*(x^2-x+1) / ((x-1)^5*(x^4+x^3+x^2+x+1)). - Colin Barker, Jun 05 2013
a(n) = floor(C(n,5)/n). - Alois P. Heinz, Jun 05 2013
G.f.: x^5/5 * (1/(1-x)^5 - 1/(1-x^5)). - Herbert Kociemba, Oct 16 2016

A011940 a(n) = floor(n(n-1)(n-2)*(n-3)/30).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 12, 28, 56, 100, 168, 264, 396, 572, 800, 1092, 1456, 1904, 2448, 3100, 3876, 4788, 5852, 7084, 8500, 10120, 11960, 14040, 16380, 19000, 21924, 25172, 28768, 32736, 37100, 41888, 47124, 52836, 59052, 65800, 73112, 81016, 89544, 98728, 108600, 119196, 130548, 142692, 155664, 169500

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A011795, A011915.

[Floor(n*(n-1)*(n-2)*(n-3)/30): n in [0..60]]; // Vincenzo Librandi, Jun 19 2012

CoefficientList[Series[4*x^5*(1-x+x^2)/((1-x)^4*(1-x^5)),{x,0,60}],x] (* Vincenzo Librandi, Jun 19 2012 *)
LinearRecurrence[{4,-6,4,-1,1,-4,6,-4,1},{0,0,0,0,0,4,12,28,56},60] (* Harvey P. Dale, Nov 13 2017 *)
Floor[4*Binomial[Range[0,60], 4]/5] (* G. C. Greubel, Oct 27 2024 *)

[4*binomial(n,4)//5 for n in range(61)] # G. C. Greubel, Oct 27 2024

a(n) = 4 * A011795(n).
From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-5) - 4*a(n-6) + 6*a(n-7) - 4*a(n-8) + a(n-9).
G.f.: 4*x^5*(1-x+x^2) / ((1-x)^5*(1+x+x^2+x^3+x^4) ). (End)

A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads.

Original entry on oeis.org

0, 1, 5, 18, 55, 143, 333, 715, 1430, 2700, 4862, 8398, 13995, 22610, 35530, 54477, 81719, 120175, 173583, 246675, 345345, 476901, 650325, 876525, 1168695, 1542684, 2017356, 2615085, 3362260, 4289780, 5433714, 6835972, 8544965, 10616463, 13114465, 16112057

Offset: 0

Criel Merino, Pedro Antonio, Oct 14 2015

nonn

A row of triangle A051168.

Pedro Antonio, Table of n, a(n) for n = 0..100
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (6, -15, 23, -33, 51, -64, 63, -63, 64, -51, 33, -23, 15, -6, 1).

Cf. A001840, A006918, A011795, A011796, A011797, A051168.

CoefficientList[Series[(x (x^4 - x^3 + 3*x^2 - x + 1))/((x^2 + x + 1)^3 (1 - x)^9), {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 15 2015 *)
CoefficientList[Series[((-1+x^3)^-3-(-1+x)^-9)/9,{x,0,40}],x] (* Herbert Kociemba, Oct 16 2016 *)
LinearRecurrence[{6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1},{0,1,5,18,55,143,333,715,1430,2700,4862,8398,13995,22610,35530},40] (* Harvey P. Dale, Feb 10 2023 *)

a(n)= (1/(n+9))*sumdiv(gcd(n+9,9), d, moebius(d)*binomial( (n+9)/d , 9/d )); \\ Michel Marcus, Oct 14 2015

from sympy import mobius, binomial, gcd, divisors
print([sum(mobius(d) * binomial((n + 9)//d, 9//d) for d in divisors(gcd(n + 9, 9))) // (n + 9) for n in range(51)]) # Indranil Ghosh, Mar 26 2017

a(n) = (1/(n+9))*Sum_{d divides gcd(n+9,9)} mu(d)*binomial((n+9)/d, 9/d).
G.f.: (x*(x^4-x^3+3*x^2-x+1))/((x^2+x+1)^3*(1-x)^9).
G.f.: ((-1+x^3)^-3-(-1+x)^-9)/9. - Herbert Kociemba, Oct 16 2016

More terms from Michel Marcus, Oct 14 2015

cp's OEIS Frontend

A051168 Triangular array T(h,k) for 0 <= k <= h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.

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A011847 Triangle of numbers read by rows: T(n,k) = floor( C(n,k)/(k+1) ), where k=0..n-1 and n >= 1.

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A011915 a(n) = floor(n*(n-1)*(n-2)*(n-3)/5).

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A124813 Number of 4-ary Lyndon words of length n with exactly five 1s.

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A051170 T(n,5), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 5 black beads and n-5 white beads.

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A011940 a(n) = floor(n*(n-1)*(n-2)*(n-3)/30).

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A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads.

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A011915 a(n) = floor(n(n-1)(n-2)*(n-3)/5).

A011940 a(n) = floor(n(n-1)(n-2)*(n-3)/30).