A051168 -id:A051168 - cp's OEIS frontend

A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

0, 0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220, 4847208, 17678835, 64821680, 238819350, 883629164, 3282060210, 12233125112, 45741281820, 171529777432, 644952073662, 2430973096720, 9183676536076

Offset: 0

N. J. A. Sloane

Number of Dyck paths of length 2n having an odd number of peaks at even height. Example: a(3)=2 because we have UDU(UD)D and U(UD)DUD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004
For n>=1, a(n) is the number of unordered binary trees with n internal nodes in which the left subtree is distinct from the right subtree. - Geoffrey Critzer, Feb 21 2013
Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - R. J. Mathar, Jul 19 2016

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.26).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n=0..200
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
Index entries for sequences related to Lyndon words

a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.
Cf. A051168, A005430.
Cf. A007595.
A diagonal of the square array described in A051168.

nn=20;CoefficientList[Series[x/2(((1-(1-4x)^(1/2))/(2x))^2-(1-(1-4x^2)^(1/2))/(2x^2)),{x,0,nn}],x]  (* Geoffrey Critzer, Feb 21 2013 *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.
G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - Emeric Deutsch, Nov 13 2004
With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - Geoffrey Critzer, Feb 21 2013
a(n) = ( 2^(n-3)/sqrt(Pi) ) * ( 4*2^n*GAMMA(n+1/2)/GAMMA(n+2) + ((-1)^n - 1)*GAMMA(n/2)/GAMMA(n/2 + 3/2) ) for n>0. - Mark van Hoeij, Nov 11 2009
a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014
a(2n) = A000108(2n) / 2; a(2n+1) = ( A000108(2n+1) - A000108(n) ) / 2. - John Bodeen, Jun 24 2015
D-finite with recurrence +n*(n+1)*(n-2)^2*a(n) -2*n*(2*n-5)*(n-1)^2*a(n-1) -4*n*(n-2)^3*a(n-2) +8*(2*n-5)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Oct 28 2021

Additional comments from Clark Kimberling

A011797 a(n) = floor(C(n,6)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 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

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) is the number of aperiodic necklaces (Lyndon words) with 7 black beads and n-7 white beads.

David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
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,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Cf. A000031, A001037, A051168. Same as A051172(n+1).
First differences of A011853.
A column of triangle A011847.

CoefficientList[Series[x^6/7 (1/(1-x)^7-1/(1- x^7)),{x,0,40}],x]; (* Herbert Kociemba, Oct 16 2016 *)

a(n) = binomial(n, 6)\7; \\ Michel Marcus, Oct 16 2016

G.f.: (1+x^3)^2/((1-x)^4(1-x^2)^2(1-x^7))*x^7.
a(n) = floor(binomial(n+1,7)/(n+1)). [Gary Detlefs, Nov 23 2011]
G.f.: (x^6/7)*(1/(1-x)^7-1/(1- x^7)). - Herbert Kociemba, Oct 16 2016

A031164 Irreducible Euler sums of weight 8 and depth 10+2n.

Original entry on oeis.org

1, 4, 15, 40, 99, 212, 429, 800, 1430, 2424, 3978, 6288, 9690, 14520, 21318, 30624, 43263, 60060, 82225, 110968, 148005, 195052, 254475, 328640, 420732, 533936, 672452, 840480, 1043460, 1286832, 1577532, 1922496, 2330445

Offset: 0

David Broadhurst

a(n-9)=number of aperiodic necklaces (Lyndon words) with 8 black beads and n-8 white beads.

David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
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 (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).

Cf. A000031, A001037, A051168. Row 8 in A245558.

Cf. A032094. - M. F. Hasler, May 02 2009

Table[(Binomial[n+8,7]-If[OddQ[n],1,0]Binomial[(n+7)/2,3])/8,{n,0,40}] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^8 (1+x)^4),{x,0,40}],x] (* Harvey P. Dale, Jun 20 2011 *)

A031164(n)=(binomial(n+8,7)-if(n%2,binomial(n\2+4,3)))>>3 \\ M. F. Hasler, May 02 2009

G.f.: (1+x^2)/((1-x)*(1-x^2))^4
a(n) = [C(n+8,7)-(n%2)*C((n+7)/2,3)]/8, where C = binomial, n%2 = parity of n (=1 if odd, 0 else). - M. F. Hasler, May 02 2009
a(0)=1, a(1)=4, a(2)=15, a(3)=40, a(4)=99, a(5)=212, a(6)=429, a(7)=800, a(8)=1430, a(9)=2424, a(10)=3978, a(11)=6288, a(n) = 4*a(n-1)-2*a(n-2)-12*a(n-3)+17*a(n-4)+8*a(n-5)-28*a(n-6)+8*a(n-7)+17*a(n-8)-12*a(n-9)- 2*a(n-10)+4*a(n-11)-a(n-12). - Harvey P. Dale, Jun 20 2011
G.f.: ((-1+x)^-8-(-1+x^2)^-4)/(8*x). - Herbert Kociemba, Oct 16 2016

A059865 Product_{i=4..n} (prime(i) - 6).

Original entry on oeis.org

1, 1, 1, 1, 5, 35, 385, 5005, 85085, 1956955, 48923875, 1516640125, 53082404375, 1964048961875, 80526007436875, 3784722349533125, 200590284525255625, 11032465648889059375, 672980404582232621875, 43743726297845120421875

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Arises in Hardy-Littlewood prime k-tuplet conjectural formulas. Also the sequence gives the exact numbers of X42424Y difference-pattern in dRRS[m], where m=modulus=A002110(n). See A049296 (=dRRS[210]=list of first differences of reduced residue system modulo 210=4th primorial). A pattern X42424Y corresponds to a residue-sextuple or it is their difference-quintuple, X,Y > 4. Analogous pattern for primes is in A022008.

a(352) has 1001 decimal digits. - Michael De Vlieger, Mar 06 2017

			a(7) = (prime(4)-6) * (prime(5)-6) * (prime(6)-6) * (prime(7)-6) = 1 * 5* 7 *11 = 385
 Also in one period of dRRS with 2,6,30,210,2310,... modulus [A002110(n)] 1,2,8,48,480,... differences occur [A005867(n)]. The number of X42424Y residue-difference-patterns are 0,1,1,1,5,... respectively starting at suitable residues coprime to A002110(n).

See A059862 for references.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Michael De Vlieger, Table of n, a(n) for n = 1..351
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = prod(k=4, n, prime(k) - 6); \\ Michel Marcus, Mar 06 2017

A124720 Number of ternary Lyndon words of length n with exactly two 1's.

Original entry on oeis.org

2, 5, 16, 38, 96, 220, 512, 1144, 2560, 5616, 12288, 26592, 57344, 122816, 262144, 556928, 1179648, 2490112, 5242880, 11009536, 23068672, 48233472, 100663296, 209713152, 436207616, 905965568, 1879048192, 3892305920, 8053063680, 16642981888, 34359738368

Offset: 3

Mike Zabrocki, Nov 05 2006

If the offsets are modified, A124720 to A124723 are the 2nd to 5th Witt transform of A000079 [Moree]. - R. J. Mathar, Nov 08 2008

a(n+2) is the number of distinct unordered pairs of binary words having a total length of n letters: a(2+2) = 5 because we have the unordered pairs: (e,00),(e,01), (e,10), (e,11), (0,1) where e represents the empty word. Each pair has a total of 2 letters and the two elements of each pair are distinct words. - Geoffrey Critzer, Feb 28 2013

			a(4) = 5 because 1122, 1123, 1132, 1213, 1133 are all Lyndon words on 3 letters with 2 ones.

Colin Barker, Table of n, a(n) for n = 3..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

Cf. A051168, A027376, A124721, A124722, A124723.

nn=30;Drop[CoefficientList[Series[(1/(1-2x)^2-1/(1-2x^2))/2,{x,0,nn}],x],1] (* Geoffrey Critzer, Feb 28 2013 *)

Vec(x^3*(2-3*x)/((1-2*x)^2*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Oct 28 2016

G.f.: x^3*(2-3 x)/((1-2 x^2)(1- 2x)^2) = (x^2/(1-2x)^2 - x^2/(1-2*x^2))/2.
From Colin Barker, Oct 28 2016: (Start)
a(n) = 2^(n-3)*(n-1)-2^(n/2-2) for n even.
a(n) = 2^(n-3)*n-2^(n-3) for n odd.
a(n) = 4*a(n-1)-2*a(n-2)-8*a(n-3)+8*a(n-4) for n>6.
(End)

A124814 Triangle of number of 4-ary Lyndon words of length n containing exactly k 1s.

Original entry on oeis.org

1, 3, 1, 3, 3, 0, 8, 9, 3, 0, 18, 27, 12, 3, 0, 48, 81, 54, 18, 3, 0, 116, 243, 198, 89, 21, 3, 0, 312, 729, 729, 405, 135, 27, 3, 0, 810, 2187, 2538, 1701, 702, 189, 30, 3, 0, 2184, 6561, 8748, 6801, 3402, 1134, 251, 36, 3, 0, 5880, 19683, 29484, 26244, 15282, 6123, 1692

Offset: 0

Mike Zabrocki, Nov 08 2006

Row sums given by A027377, first column given by A027376, second column given by A000244, third through sixth columns (k=2,3,4,5) given by A124810, A124811, A124812, A124813, third diagonal given by 3*A032766.

			T(4,2) = 12 because the words 11ab, 11ba, 1a1b for ab=23, 24, 34 and 11aa for a=2,3,4 are all Lyndon and of length 4 with exactly two 1s.
From _Andrew Howroyd_, Mar 26 2017: (Start)
Triangle starts
*   1
*   3    1
*   3    3    0
*   8    9    3    0
*  18   27   12    3   0
*  48   81   54   18   3   0
* 116  243  198   89  21   3  0
* 312  729  729  405 135  27  3 0
* 810 2187 2538 1701 702 189 30 3 0
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Cf. A051168, A123223, A027377, A124810, A124811, A124812, A124813.

C:=combinat[numbcomb]:mu:=numtheory[mobius]:divs:=numtheory[divisors]: T:=proc(n,k) local d; if k>0 then add(mu(d)*C(n/d-1,(n-k)/d)*3^((n-k)/d),d=divs(n) intersect divs(k))/k; elif n>0 then 1/n*add(mu(d)*3^(n/d),d=divs(n)); else 1; fi; end; [seq([seq(T(n,k),k=0..n)],n=0..10)];

nmax = 10; col[0] = Table[If[n == 0, 1, 1/n* DivisorSum[n, MoebiusMu[#]* 3^(n/#)&]], {n, 0, nmax}]; col[k_] := x^k/k * DivisorSum[k, MoebiusMu[#] / (1 - 3*x^#)^(k/#)&] + O[x]^(nmax+2) // CoefficientList[#, x]&; Table[ col[k][[n+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2017 *)

T(n,0) = 1/n*Sum_{d|n} mu(d)*3^(n/d) = A027376(n).
T(n,n-1) = 3 for k>0.
T(n,k) = 1/k*Sum_{d|k,d|n} mu(d) C(n/d-1,(n-k)/d )*3^((n-k)/d) = 1/(n-k)*Sum_{d|k,d|n} mu(d) C(n/d-1,k/d)*3^((n-k)/d).
O.g.f. of columns: Sum_n T(n,k) x^n = x^k/k*Sum_{d|k} mu(d)*1/(1-3*x^d)^(k/d).
O.g.f. of diagonals: Sum_n T(n,n-k) x^n = x^k/k*Sum_{d|k} mu(d)*(3/(1-x^d))^(k/d).

A124721 Number of ternary Lyndon words with exactly three 1's.

Original entry on oeis.org

2, 8, 26, 80, 224, 596, 1536, 3840, 9384, 22528, 53248, 124240, 286720, 655360, 1485472, 3342336, 7471104, 16602432, 36700160, 80740352, 176859776, 385875968, 838860800, 1817531648, 3925868544, 8455716864, 18164132352, 38923141120

Offset: 4

Mike Zabrocki, Nov 05 2006

			a(5) = 8 because 11122, 11212, 11123, 11132, 11213, 11312, 11133, 11313 are all ternary Lyndon words of length 5 with three 1's

Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-12,24,-16).

Cf. A051168, A027376, A124720, A124722, A124723.

G.f.: 2*x^4*(x - 1)^2/(1-2*x^3)/(1-2*x)^3 = (x^3/(1-2*x)^3-x^3/(1-2*x^3))/3

A124722 Number of ternary Lyndon words with exactly four 1's.

Original entry on oeis.org

2, 9, 40, 137, 448, 1336, 3840, 10540, 28160, 73168, 186368, 465808, 1146880, 2785024, 6684672, 15875520, 37355520, 87161600, 201850880, 464254208, 1061158912, 2411718656, 5452595200, 12268325888, 27481079808, 61303918592

Offset: 5

Mike Zabrocki, Nov 05 2006

			a(6) = 9 because 111122, 111212, 111123, 111213, 112113, 111132, 111312, 111133, 111313 are all ternary Lyndon words with four 1's

Index entries for linear recurrences with constant coefficients, signature (8,-20,0,76,-96,-32,128,-64).

Cf. A051168, A027376, A124720, A124721, A124723.

LinearRecurrence[{8,-20,0,76,-96,-32,128,-64},{2,9,40,137,448,1336,3840,10540},40] (* Harvey P. Dale, Nov 04 2020 *)

G.f.: x^5*(2-3*x)*(1-x)^2/(1 - 2*x^2)^2/(1 - 2*x)^4 = (1/(1-2*x)^4-1/(1-2*x^2)^2)/4

A124723 Number of ternary Lyndon words with exactly five 1's.

Original entry on oeis.org

2, 12, 56, 224, 806, 2688, 8448, 25344, 73216, 205004, 559104, 1490944, 3899392, 10027008, 25401752, 63504384, 156893184, 383516672, 928514048, 2228433712, 5305794560, 12540968960, 29444014080, 68702699520, 159390262880

Offset: 6

Mike Zabrocki, Nov 05 2006

nonn

			a(7) = 12 because 11111ab, 1111a1b, 111a11b where ab = 22, 23, 32 or 33 are all ternary Lyndon words of length 7 with five 1's.

Cf. A051168, A027376, A124720, A124721, A124722.

a:= n-> (Matrix([[806, 224, 56, 12, 2, 0$5]]). Matrix(10, (i,j)-> `if`(i=j-1, 1, `if`(j=1, [10, -40, 80, -80, 34, -20, 80, -160, 160, -64] [i], 0)))^(n-10))[1,1]: seq(a(n), n=6..30);  # Alois P. Heinz, Aug 04 2008

G.f.: 2*x^6*(1-2*x+3*x^2)*(1-x)^2/(1-2*x^5)/(1-2*x)^5= (1/(1-2*x)^5-1/(1-2*x^5))/5.

A059862 a(n) = Product_{i=3..n} (prime(i) - 3).

Original entry on oeis.org

1, 1, 2, 8, 64, 640, 8960, 143360, 2867200, 74547200, 2087321600, 70968934400, 2696819507200, 107872780288000, 4746402332672000, 237320116633600000, 13289926531481600000, 770815738825932800000, 49332207284859699200000, 3354590095370459545600000, 234821306675932168192000000

Offset: 1

Labos Elemer, Feb 28 2001

nonn

			For n = 6, a(6) = 640 because:
prime(1..6)-3 = (-1,0,2,4,8,10) -> (1,1,2,4,8,10)
and
1*1*2*4*8*10 = 640. [Example generalized and reformatted per observation of _Jon E. Schoenfield_ by _Harlan J. Brothers_, Jul 15 2018]

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly, 66 (1959), 375-384.

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

a:= proc(n) option remember;
      `if`(n<3, 1, a(n-1)*(ithprime(n)-3))
    end:
seq(a(n), n=1..21);  # Alois P. Heinz, Nov 19 2021

Join[{1, 1}, Table[Product[Prime[i] - 3, {i, 3, n}], {n, 3, 19}]] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 3);
Table[a[n], {n, 19}] (* Harlan J. Brothers, Jul 02 2018 *)

a(n) = prod(i=3, n, prime(i) - 3); \\ Michel Marcus, Jul 15 2018

a(1) = a(2) = 1; a(n) = a(n-1) * (prime(n) - 3) for n >= 3. - David A. Corneth, Jul 15 2018

Name clarified, offset corrected by David A. Corneth, Jul 15 2018

cp's OEIS Frontend

A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

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A011797 a(n) = floor(C(n,6)/7).

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A031164 Irreducible Euler sums of weight 8 and depth 10+2n.

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A059865 Product_{i=4..n} (prime(i) - 6).

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A124720 Number of ternary Lyndon words of length n with exactly two 1's.

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A124814 Triangle of number of 4-ary Lyndon words of length n containing exactly k 1s.

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A124721 Number of ternary Lyndon words with exactly three 1's.

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A124722 Number of ternary Lyndon words with exactly four 1's.