A001006 -id:A001006 - cp's OEIS frontend

A266969 Integers k such that A001006(k) is divisible by k.

1, 2, 21, 266, 387, 657, 1314, 6291, 16113, 25767, 27594, 41902, 54243, 55314, 56457, 89018, 96141, 155601, 172746, 219842, 294273, 300871, 384426, 412398, 453781, 579474, 653421, 660879, 669609, 951881, 993307, 1117338, 1246077, 1401258, 1438623, 1535409, 1870533

Offset: 1

Altug Alkan, Jan 07 2016

nonn

Integers n such that number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle is divisible by n.

Corresponding values of A001006(n)/n are 1, 1, 6787979, ...

			There are 142547559 ways to join 21 points on a circle by nonintersecting chords. Because of the fact that 142547559 is divisible by 21, 21 is a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..70

Cf. A001006, A081735.

lim = 100000; t = CoefficientList[Series[(1 - x - (1 - 2 x - 3 x^2)^(1/2))/(2 x^2), {x, 0, lim}], x]; Select[Range@ lim, Divisible[t[[# + 1]], #] &] (* Michael De Vlieger, Jan 09 2016, after Jean-François Alcover at A001006 *)
seq[kmax_] := Module[{mot1 = 1, mot2 = 2, mot, s = {1, 2}}, Do[mot3 = ((2*k+1)*mot2 + (3*k-3)*mot1)/(k+2); If[Divisible[mot3, k], AppendTo[s, k]]; mot1 = mot2; mot2 = mot3, {k, 3, kmax}]; s]; seq[10^5] (* Amiram Eldar, May 12 2024 *)

lista(kmax) = {my(mot1 = 1, mot2 = 2, mot); print1(1, ", ", 2, ", "); for(k = 3, kmax, mot3 = ((2*k+1)*mot2 + (3*k-3)*mot1)/(k+2); if(!(mot3 % k), print1(k,", ")); mot1 = mot2; mot2 = mot3);} \\ Amiram Eldar, May 12 2024

a(8)-a(17) from Michael De Vlieger, Jan 09 2016

a(18)-a(37) from Amiram Eldar, May 12 2024

A114422 Riordan array (1/sqrt(1-2x-3x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 9, 5, 1, 19, 26, 19, 7, 1, 51, 75, 65, 33, 9, 1, 141, 216, 211, 132, 51, 11, 1, 393, 623, 665, 483, 235, 73, 13, 1, 1107, 1800, 2058, 1674, 963, 382, 99, 15, 1, 3139, 5211, 6294, 5598, 3663, 1739, 581, 129, 17, 1, 8953, 15115, 19095, 18261, 13243

Offset: 0

Paul Barry, Feb 12 2006

First column is central trinomial numbers A002426.
Second column is A005774.
Third column is A025568.
Row sums are A116387.
Diagonal sums are A116388.
Product of A007318 and A116382.
Column k has e.g.f. exp(x)*Sum_{j=0..k} C(k,j)*Bessel_I(k+j,2*x).

			Triangle begins
1,
1, 1,
3, 3, 1,
7, 9, 5, 1,
19, 26, 19, 7, 1,
51, 75, 65, 33, 9, 1,
141, 216, 211, 132, 51, 11, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j))))); # G. C. Greubel, Dec 15 2018

[[(&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018

T[n_, k_] := Sum[Binomial[n, j - k]*Binomial[j, n - j], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

{T(n,k) = sum(j=0,n, binomial(n, j-k)*binomial(j, n-j))};
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2018

[[sum(binomial(n, j-k)*binomial(j, n-j) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018

Riordan array (1/sqrt(1-2*x-3*x^2), (1-x-2*x^2-sqrt(1-2*x-3*x^2) ) / (2*x^2)).

Number triangle T(n,k) = Sum_{j=0..n} C(n,j-k)*C(j,n-j).

A039964 Motzkin numbers A001006 read mod 3.

Original entry on oeis.org

1, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

An example of a d-perfect sequence.

The asymptotic mean of this sequence is 0 (Burns, 2016). - Amiram Eldar, Jan 30 2021

Amiram Eldar, Table of n, a(n) for n = 0..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985.
David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.

Cf. A001006.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

b = DifferenceRoot[Function[{b, n}, {3 (n + 1) b[n] + (2 n + 5) b[n + 1] == (n + 4) b[n + 2], b[0] == 1, b[1] == 1}]];
a[n_] := Mod[b[n], 3];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 26 2019 *)

a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);
vector(200, n, n--; a001006(n) % 3) \\ Altug Alkan, Oct 23 2015

a(n) = A001006(n) mod 3. - Christian G. Bower, Jun 12 2005

More terms from Christian G. Bower, Jun 12 2005

Offset adapted by Altug Alkan, Oct 23 2015

A258710 Motzkin numbers A001006 read mod 11.

Original entry on oeis.org

1, 1, 2, 4, 9, 10, 7, 6, 4, 10, 10, 1, 1, 2, 4, 9, 10, 7, 6, 4, 9, 0, 3, 3, 6, 1, 5, 8, 10, 7, 1, 6, 10, 7, 7, 3, 6, 8, 4, 5, 9, 6, 9, 10, 8, 8, 5, 10, 6, 3, 1, 4, 10, 9, 8, 7, 7, 3, 6, 8, 4, 5, 9, 6, 3, 5, 9, 9, 7, 3, 4, 2, 8, 10, 3, 8, 7, 8, 8, 5, 10, 6, 3, 1, 4, 10

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Rob Burns, Structure and asymptotics for Motzkin numbers modulo small primes using automata, arXiv:1612.08146 [math.NT], 2016.
Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985

Cf. A001006.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

m[0] = 1; m[n_] := m[n] = m[n-1] + Sum[m[k] m[n-k-2], {k, 0, n-2}];
a[n_] := Mod[m[n], 11];
Array[a, 100, 0] (* Jean-François Alcover, Nov 05 2018 *)

A258711 Motzkin numbers A001006 read mod 7.

Original entry on oeis.org

1, 1, 2, 4, 2, 0, 2, 1, 1, 2, 4, 2, 6, 3, 3, 3, 6, 5, 6, 5, 1, 0, 0, 0, 0, 0, 1, 6, 5, 5, 3, 6, 3, 5, 5, 2, 2, 4, 1, 4, 4, 0, 1, 1, 2, 4, 2, 0, 2, 1, 1, 2, 4, 2, 0, 2, 1, 1, 2, 4, 2, 6, 3, 3, 3, 6, 5, 6, 5, 1, 0, 0, 0, 0, 0, 1, 6, 5, 5, 3, 6, 3, 5, 5, 2, 2, 4, 1, 4

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Rob Burns, Structure and asymptotics for Motzkin numbers modulo small primes using automata, arXiv:1612.08146 [math.NT], 2016.
Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985

Cf. A001006.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

m[0] = 1; m[n_] := m[n] = m[n-1] + Sum[m[k] m[n-k-2], {k, 0, n-2}];
a[n_] := Mod[m[n], 7];
Array[a, 100, 0] (* Jean-François Alcover, Nov 05 2018 *)

a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);
vector(200, n, n--; a001006(n) % 7) \\ Altug Alkan, Oct 23 2015

A258712 Motzkin numbers A001006 read mod 5.

Original entry on oeis.org

1, 1, 2, 4, 4, 1, 1, 2, 3, 0, 3, 3, 1, 0, 4, 2, 2, 4, 2, 4, 4, 4, 3, 0, 2, 1, 1, 2, 4, 4, 1, 1, 2, 3, 0, 3, 3, 1, 0, 4, 2, 2, 4, 2, 4, 4, 4, 3, 4, 3, 3, 3, 1, 2, 2, 3, 3, 1, 4, 0, 4, 4, 3, 0, 2, 1, 1, 2, 1, 2, 2, 2, 4, 3, 3, 2, 2, 4, 3, 3, 2, 2, 4, 1, 0, 1, 1, 2, 0

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985

Cf. A001006.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

b = DifferenceRoot[Function[{b, n}, {3 (n + 1) b[n] + (2 n + 5) b[n + 1] == (n + 4) b[n + 2], b[0] == 1, b[1] == 1}]];
a[n_] := Mod[b[n], 5];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2019 *)

a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);
vector(200, n, n--; a001006(n) % 5) \\ Altug Alkan, Oct 23 2015

A299918 Motzkin numbers (A001006) mod 8.

Original entry on oeis.org

1, 1, 2, 4, 1, 5, 3, 7, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 7, 7, 5, 5, 4, 2, 1, 5, 3, 7, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 4, 6, 5, 5, 2, 4, 5, 1, 1, 1, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 6, 4, 1, 1, 2, 4, 1

Offset: 0

N. J. A. Sloane, Mar 16 2018

Robert Israel, Table of n, a(n) for n = 0..10000
S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Catalan and Motzkin numbers modulo 4 and 8, Europ. J. Combin. 29 (2008), 1449-1466.
Christian Krattenthaler and Thomas W. Müller, Motzkin numbers and related sequences modulo powers of 2, arXiv:1608.05657 [math.CO], 2016-2018.
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, J. Théorie Nombres Bordeaux 27 (2015), 245-288.
Ying Wang and Guoce Xin, A Classification of Motzkin Numbers Modulo 8, Electron. J. Combin., 25(1) (2018), #P1.54.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

f:= rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1}, a(n), remember): seq(f(n) mod 8, n=0..200); # Robert Israel, Mar 16 2018

Table[Mod[GegenbauerC[n, -n - 1, -1/2] / (n + 1), 8], {n, 0, 100}] (* Vincenzo Librandi, Sep 08 2018 *)

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1)) % 8; \\ Michel Marcus, May 23 2022

A299919 Motzkin numbers (A001006) mod 4.

Original entry on oeis.org

1, 1, 2, 0, 1, 1, 3, 3, 3, 3, 0, 2, 3, 3, 2, 0, 3, 3, 2, 0, 3, 3, 3, 3, 1, 1, 0, 2, 1, 1, 3, 3, 3, 3, 2, 0, 3, 3, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 1, 1, 3, 3, 0, 2, 3, 3, 2, 0, 3, 3, 2, 0, 3, 3, 1, 1, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 2, 0, 1

Offset: 0

N. J. A. Sloane, Mar 16 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

f:= rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1},a(n),remember):
seq(f(n) mod 4, n=0..200); # Robert Israel, Mar 16 2018

b = DifferenceRoot[Function[{b, n}, {3(n+1) b[n] + (2n+5) b[n+1] == (n+4) b[n+2], b[0] == 1, b[1] == 1}]];
a[n_] := Mod[b[n], 4];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2019 *)

A299920 Motzkin numbers (A001006) mod 6.

Original entry on oeis.org

1, 1, 2, 4, 3, 3, 3, 1, 5, 1, 4, 2, 1, 3, 0, 0, 3, 3, 0, 0, 3, 3, 3, 3, 3, 1, 2, 4, 1, 5, 1, 3, 3, 3, 4, 2, 1, 1, 5, 1, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 3, 3, 0, 4, 5, 1, 4, 2, 1, 3

Offset: 0

N. J. A. Sloane, Mar 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

f:= gfun:-rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1},a(n),remember):
seq(f(n) mod 6, n=0..100); # Robert Israel, Mar 16 2018

b = DifferenceRoot[Function[{b, n}, {3 (n + 1) b[n] + (2 n + 5) b[n + 1] == (n + 4) b[n + 2], b[0] == 1, b[1] == 1}]];
a[n_] := Mod[b[n], 6];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2019 *)

A187306 Alternating sum of Motzkin numbers A001006.

Original entry on oeis.org

1, 0, 2, 2, 7, 14, 37, 90, 233, 602, 1586, 4212, 11299, 30536, 83098, 227474, 625993, 1730786, 4805596, 13393688, 37458331, 105089228, 295673995, 834086420, 2358641377, 6684761124, 18985057352, 54022715450, 154000562759, 439742222070, 1257643249141

Offset: 0

Paul Barry, Mar 08 2011

Diagonal sums of A089942.
Hankel transform is A187307.
Also gives the number of simple permutations of each length that avoid the pattern 321 (i.e., are the union of two increasing sequences, and in one line notation contain no nontrivial block of values which form an interval). There are 2 such permutations of length 4, 2 of length 5, etc. - Michael Albert, Jun 20 2012
Convolution of A005043 with itself. - Philippe Deléham, Jan 28 2014
From Gus Wiseman, Nov 15 2022: (Start)
Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 2 vertices. This would imply a(n) = A284778(n-1) + A005043(n). For example, the a(0) = 1 through a(5) = 14 trees are:
  (o)  .  (ooo)   (oooo)   (ooooo)    (oooooo)
          ((oo))  ((ooo))  ((oo)oo)   ((oo)ooo)
                           ((oooo))   ((ooo)oo)
                           (o(oo)o)   ((ooooo))
                           (oo(oo))   (o(oo)oo)
                           (((oo)o))  (o(ooo)o)
                           ((o(oo)))  (oo(oo)o)
                                      (oo(ooo))
                                      (ooo(oo))
                                      (((oo)oo))
                                      (((ooo)o))
                                      ((o(oo)o))
                                      ((o(ooo)))
                                      ((oo(oo)))
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
M. H. Albert and V. Vatter, Generating and enumerating 321-avoiding and skew-merged simple permutations, arXiv preprint arXiv:1301.3122 [math.CO], 2013. - _N. J. A. Sloane_, Feb 11 2013
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.

Cf. A001006, A005043, A089942.

a := n -> (-1)^n*(1-hypergeom([1/2,-n-1],[2],4));
seq(round(evalf(a(n),99)),n=0..30); # Peter Luschny, Sep 25 2014

CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(2x^2(1+x)), {x,0,30}], x] (* Harvey P. Dale, Jun 14 2011 *)

x='x+O('x^66); Vec((1-x-sqrt(1-2*x-3*x^2))/(2*x^2*(1+x))) /* Joerg Arndt, Mar 07 2013 */

Vec(serreverse(x*(1-x)/(1-x+x^2) + O(x^30))^2) \\ Andrew Howroyd, Apr 28 2018

def A187306():
    a, b, n = 1, 0, 1
    yield a
    while True:
        n += 1
        a, b = b, (2*b+3*a)*(n-1)/(n+1)
        yield b - (-1)^n
A187306_list = A187306()
[next(A187306_list) for i in range(20)] # Peter Luschny, Sep 25 2014

G.f.: (1-x-sqrt(1-2*x-3*x^2))/(2*x^2*(1+x)).
a(n) = sum(k=0..n, A001006(k)*(-1)^(n-k)).
D-finite with recurrence -(n+2)*a(n) +(n-1)*a(n-1) +(5*n-2)*a(n-2) +3*(n-1)a(n-3)=0. - R. J. Mathar, Nov 17 2011
a(n) = (2*sum(j=0..n, C(2*j+1,j+1)*(-1)^(n-j)*C(n+2,j+2)))/(n+2). - Vladimir Kruchinin, Feb 06 2013
a(n) ~ 3^(n+5/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 15 2013
a(n) = (-1)^n*(1-hypergeom([1/2,-n-1],[2],4)). - Peter Luschny, Sep 25 2014
a(n) = A005043(n+1) + (-1)^n. - Peter Luschny, Sep 25 2014
G.f.: (1/(1 - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))))^2, a continued fraction. - Ilya Gutkovskiy, Sep 23 2017

cp's OEIS Frontend

A266969 Integers k such that A001006(k) is divisible by k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A114422 Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

Views

Author

Keywords

Comments

Examples

Links

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A039964 Motzkin numbers A001006 read mod 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A258710 Motzkin numbers A001006 read mod 11.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A258711 Motzkin numbers A001006 read mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A258712 Motzkin numbers A001006 read mod 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A299918 Motzkin numbers (A001006) mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A299919 Motzkin numbers (A001006) mod 4.

Views

Author

Keywords

Links

A114422 Riordan array (1/sqrt(1-2x-3x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.