A051168 -id:A051168 - cp's OEIS frontend

A092964 Numbers > 1 in A051168, with a(0) = 1.

1, 2, 2, 2, 3, 2, 3, 5, 5, 3, 3, 7, 8, 7, 3, 4, 9, 14, 14, 9, 4, 4, 12, 20, 25, 20, 12, 4, 5, 15, 30, 42, 42, 30, 15, 5, 5, 18, 40, 66, 75, 66, 40, 18, 5, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 6, 26, 70, 143, 212, 245, 212, 143, 70, 26, 6, 7, 30, 91, 200, 333, 429, 429, 333, 200

Offset: 0

Thomas O. Hoffbauer (Thomas.Hoffbauer(AT)cibamberg.de), Apr 20 2004

Original definition: Triangle read by rows in which row n gives the number of equal configurations under cyclic shift.
T(n,k) = A051168(n+3,k+1), if 0Michael Somos, Jul 17 2004
From Paul Weisenhorn, Dec 21 2010 (Start):
T(n,k) is the number of classes that have (k+1) ordered sums of (n+4) with (k+1) positive integers, that can be transformed into each other by a cyclic permutation.
m has 2^(m-1) ordered sums; for each sum one remove the first part z(1) and add 1 to the next z(1) parts to get a new ordered sum until a period is reached. T(n,k)=a(m) with m=(n+4)*(n+3)/2+k+1 gives for m the number of periods with length (n+4).
The numbers m=n*(n+3)/2 with 1<=n have one period with length (n+1).
The numbers m=n*(n+3)/2+2 with 1<=n have one period with length (n+2).
The triangular numbers n*(n+1)/2 with 1<=n have one period [(n+(n-1)+...+2+1)] with length 1. (End)

			As triangle, starts:
  1;
  2,2;
  2,3,2;
  3,5,5,3;
  3,7,8,7,3;
  4,9,14,14,9,4;
  4,12,20,25,20,12,4; ...
From _Paul Weisenhorn_, Dec 21 2010: (Start)
T(2,2)=3 classes with 3 ordered sums of 6; [(1+1+4),(1+4+1),(4+1+1)]; [(1+2+3),(2+3+1),(3+1+2)]; [(1+3+2),(3+2+1),(2+1+3)].
T(2,2)=a(m)=3 periods with length 6 for m=6*5/2+3=18 [(5+5+4+3+1),(6+5+4+2+1),(6+5+3+2+1+1),(6+4+3+2+2+1),(5+4+3+3+2+1),(5+4+4+3+2)]; [(5+5+3+3+2),(6+4+4+3+1),(5+5+4+2+1+1),(6+5+3+2+2),(6+4+3+3+1+1),(5+4+4+2+2+1)]; [(5+5+3+2+2+1),(6+4+3+3+2),(5+4+4+3+1+1),(5+5+4+2+2),(6+5+3+3+1),(6+4+4+2+1+1)]. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

Row sums give A093210. Essentially the same as A051168. See A185158 for another version.

T[n_, k_] := DivisorSum[GCD[k + 1, n + 4], Binomial[(n + 4)/#, (k + 1)/#] * MoebiusMu[#] & ]/(n + 4); Table[T[n, k], {n, 0, 12}, {k, 1, n + 1}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)

T(n,k)=local(A,ps,c); n+=3; k++; if(k<1||k>=n-1, 0, 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 17 2004 */

Edited with better definition by Omar E. Pol, Jan 05 2009

A050181 T(2n+3, n), array T as in A051168; a count of Lyndon words.

Original entry on oeis.org

0, 1, 3, 9, 30, 99, 333, 1144, 3978, 13995, 49742, 178296, 643842, 2340135, 8554275, 31429026, 115997970, 429874830, 1598952366, 5967382200, 22338765540, 83859016527, 315614844558, 1190680751376, 4501802223090, 17055399281284

Offset: 0

Clark Kimberling

nonn

Index entries for sequences related to Lyndon words

Cf. A003441.

A diagonal of the square array described in A051168.

A050181 := proc(n)
    A051168(2*n+3,n) ;
end proc: # R. J. Mathar, Jul 20 2016

a[n_] := (1/(2n+3)) Sum[MoebiusMu[d] Binomial[(2n+3)/d, n/d], {d, Divisors[ GCD[n, 3]]}];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 17 2019, from PARI *)

a(n) = (1/(2*n+3))*sumdiv(gcd(n,3), d, moebius(d)*binomial((2*n+3)/d, n/d)); \\ Michel Marcus, Nov 18 2017

Conjecture: -(n-1)*(n+3)*(n+2)*a(n) + 2*(3*n-4)*(n+2)*(n+1)*a(n-1) - 4*n*(n+1)*(2*n-5)*a(n-2) + 2*(n-1)*(n+2)*(2*n-3)*a(n-3) - 4*(2*n-5)*(3*n-4)*(n+1)*a(n-4) + 8*n*(2*n-5)*(2*n-7)*a(n-5) = 0. - R. J. Mathar, Jul 20 2016
From Petros Hadjicostas, Nov 16 2017: (Start)
a(n) = (1/(2*n+3))*Sum_{d|gcd(n,3)} mu(d)*binomial((2*n+3)/d, n/d). (This is a special case of A. Howroyd's formula for double array A051168.)
a(n) = (1/(2*n+3))*(binomial(2*n+3, n) - binomial((2*n/3)+1, n/3)) if 3|n; = (1/(2*n+3))*binomial(2*n+3, n) otherwise.
Using the above formulae, one can verify R. J. Mathar's conjecture above.
(End)

A050182 a(n) = T(2*n+4, n), array T as in A051168 (a count of Lyndon words).

Original entry on oeis.org

0, 1, 3, 12, 40, 143, 497, 1768, 6288, 22610, 81686, 297160, 1086384, 3991995, 14732005, 54587280, 202995232, 757398510, 2834502346, 10637507400, 40023606896, 150946230006, 570534474698, 2160865067312, 8199711007200

Offset: 0

Clark Kimberling

nonn

We have A051168(n,k) = (1/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d, k/d) for 0 <= k <= n with n > 0. If n is odd, gcd(2*n + 4, n) = 1. If n is even, gcd(2*n + 4, n) = 2 or 4, but mu(2) = -1 and mu(4) = 0. From these facts, we can prove the formula below. - Petros Hadjicostas, Jul 27 2020

Index entries for sequences related to Lyndon words

A diagonal of the square array described in A051168.

A050182 := proc(n)
    binomial(2*n+4,n) ;
    if type(n,'even') then
        %-binomial(n+2,n/2) ;
    end if;
    %/(2*n+4) ;
end proc:
seq(A050182(n),n=0..40) ; # R. J. Mathar, Oct 28 2021

a(n) = binomial(2*n + 4, n)/(2*n + 4), if n is odd, and a(n) = (binomial(2*n + 4, n) - binomial(n + 2, n/2))/(2*n + 4), if n is even. - Petros Hadjicostas, Jul 27 2020

D-finite with recurrence -(n+4) *(n+3) *(11*n-8)*a(n) +10 *(n+3) *(7*n^2+6*n-10) *a(n-1) -60 *n *(n^2-n-5)*a(n-2) -40 *n *(7*n^2+6*n-10) *a(n-3) +16*(n-1) *(13*n+4) *(2*n-3) *a(n-4)=0. - R. J. Mathar, Oct 28 2021

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

A050183 T(2n+5,n), array T as in A051168; a count of Lyndon words.

Original entry on oeis.org

0, 1, 4, 15, 55, 200, 728, 2652, 9690, 35530, 130750, 482885, 1789515, 6653325, 24812400, 92798375, 347993910, 1308233790, 4929576600, 18615637950, 70441574000, 267058714626, 1014283603024, 3858687620200, 14702930414900

Offset: 0

Clark Kimberling

nonn

Index entries for sequences related to Lyndon words

A diagonal of the square array described in A051168.

A050183 := proc(n)
    binomial(2*n+5,n) ;
    if modp(n,5) = 0 then
        %-binomial(2*n/5+1,n/5) ;
    end if;
    %/(2*n+5) ;
end proc:
seq(A050183(n),n=0..40) ; # R. J. Mathar, Oct 28 2021

a(n) = (1/(2*n+5))*sumdiv(gcd(n,5), d, moebius(d)*binomial((2*n+5)/d, n/d)); \\ Michel Marcus, Dec 05 2017

From Petros Hadjicostas, Dec 03 2017: (Start)
a(n) = (1/(2*n+5))*Sum_{d|gcd(n,5)} mu(d)*binomial((2*n+5)/d, n/d). (This is a special case of A. Howroyd's formula for double array A051168.)
a(n) = (1/(2*n+5))*(binomial(2*n+5, n) - binomial((2*n/5)+1, n/5)) if 5|n; = (1/(2*n+5))*binomial(2*n+5, n) otherwise.
(End)

A050185 T(2n+7,n), array T as in A051168; a count of Lyndon words.

Original entry on oeis.org

0, 1, 5, 22, 91, 364, 1428, 5537, 21318, 81719, 312455, 1193010, 4552275, 17368680, 66284554, 253086480, 966955410, 3697182450, 14147884842, 54185826156, 207712333598, 796937116661, 3060338457400, 11762344331920

Offset: 0

Clark Kimberling

nonn

Is a(n) = A000588(n+3)/7 if n is not a multiple of 7? R. J. Mathar, Jul 24 2012

Index entries for sequences related to Lyndon words

A050184 T(2n+6,n), array T as in A051168; a count of Lyndon words.

Original entry on oeis.org

0, 1, 4, 18, 70, 273, 1026, 3876, 14520, 54477, 204248, 766935, 2882934, 10855425, 40939888, 154663938, 585260496, 2218309470, 8421352956, 32018897274, 121918083980, 464879984481, 1774996216144, 6785967883800

Offset: 0

Clark Kimberling

nonn

Index entries for sequences related to Lyndon words

A051172 T(n,7), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 7 black beads and n-7 white beads.

Original entry on oeis.org

0, 1, 4, 12, 30, 66, 132, 245, 429, 715, 1144, 1768, 2652, 3876, 5537, 7752, 10659, 14421, 19228, 25300, 32890, 42287, 53820, 67860, 84825, 105183, 129456, 158224, 192129, 231880, 278256, 332112, 394383, 466089, 548340

Offset: 7

Clark Kimberling

nonn

Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Cf. A000031, A001037, A051168. Same as A011797(n-1).

Drop[CoefficientList[Series[-(x^7/7)*(1/(1-x^7)+1/(-1+x)^7),{x,0,50}],x],7] (* Harvey P. Dale, Aug 04 2024 *)

G.f.: -(x^7/7)*(1/(1- x^7)+1/(-1+x)^7). - Herbert Kociemba, Oct 16 2016

a(20) corrected by Herbert Kociemba, Oct 16 2016

A052314 Triangle: Matrix square of A051168.

Original entry on oeis.org

1, 2, 1, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 6, 3, 1, 0, 0, 1, 9, 7, 4, 1, 0, 0, 1, 18, 18, 14, 5, 1, 0, 0, 1, 30, 38, 36, 18, 6, 1, 0, 0, 1, 56, 84, 96, 60, 28, 7, 1, 0, 0, 1, 99, 174, 227, 171, 98, 33, 8, 1, 0, 0, 1, 186, 372, 552, 486, 332, 140, 47, 9, 1, 0, 0, 1, 335, 755

Offset: 0

Christian G. Bower, Dec 15 1999

			1; 2,1; 1,1,0; 1,2,0,0; 1,3,1,0,0; ...

Index entries for sequences related to Lyndon words

A130513 Subtriangle of triangle in A051168: remove central column of A051168 and all columns to the right; now read by upwards diagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 2, 1, 0, 14, 7, 3, 1, 0, 42, 20, 9, 3, 1, 0, 132, 66, 30, 12, 4, 1, 0, 429, 212, 99, 40, 15, 4, 1, 0, 1430, 715, 333, 143, 55, 18, 5, 1, 0, 4862, 2424, 1144, 497, 200, 70, 22, 5, 1, 0, 16796, 8398, 3978, 1768, 728, 273, 91, 26, 6, 1, 0, 58786, 29372, 13995

Offset: 1

Philippe Deléham, Aug 08 2007

			Triangle T(n,k), 1<=k<=n, begins:
1;
1, 0;
2, 1, 0;
5, 2, 1, 0;
14, 7, 3, 1, 0;
42, 20, 9, 3, 1, 0;
132, 66, 30, 12, 4, 1, 0;
429, 212, 99, 40, 15, 4, 1, 0;

A. Errera, Analysis situs: Un problème d'énumération, Memoires Acad. Bruxelles (1931), Serie 2, Vol. 11, No. 6, 26pp.

Cf. A000108, A000150, A050181, A050182, A050183, A050184, A050185.

Table[1/(2n-k) Plus@@ (MoebiusMu[ # ]Binomial[(2n-k)/#,(n-k)/# ]&/@ Divisors[GCD[2n-k,n-k]]),{n,12},{k,n}] (* Wouter Meeussen, Jul 20 2008 *)

Sum_{k, 1<=k<=n} T(n,k) = A022553(n); Sum_{k, 1<=k<=n}k*T(n,k) = A002996(n).

T(n,k) = 1/(2n-k) Sum( d | gcd(2n-k,n-k) = mu(d) C((2n-k)/d,(n-k)/d) ). - Wouter Meeussen, Jul 20 2008

Edited by N. J. A. Sloane, Oct 08 2007

cp's OEIS Frontend

A092964 Numbers > 1 in A051168, with a(0) = 1.

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A050181 T(2n+3, n), array T as in A051168; a count of Lyndon words.

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A050182 a(n) = T(2*n+4, n), array T as in A051168 (a count of Lyndon words).

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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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A050183 T(2n+5,n), array T as in A051168; a count of Lyndon words.

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A050185 T(2n+7,n), array T as in A051168; a count of Lyndon words.

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A050184 T(2n+6,n), array T as in A051168; a count of Lyndon words.

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A051172 T(n,7), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 7 black beads and n-7 white beads.

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A052314 Triangle: Matrix square of A051168.

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