A001006 -id:A001006 - cp's OEIS frontend

A340395 a(n) = A340131(A001006(n)).

5, 15, 50, 150, 455, 1365, 4100, 12300, 36905, 110715, 332150, 996450, 2989355, 8968065, 26904200, 80712600, 242137805, 726413415, 2179240250, 6537720750, 19613162255, 58839486765, 176518460300, 529555380900, 1588666142705, 4765998428115, 14297995284350

Offset: 2

Gennady Eremin, Jan 06 2021

This sequence is associated with A340131, whose terms are sorted by the length of their ternary code. Elements with the same length of ternary code form a range that has a maximum. The maximal term of the n-range (a set of elements with ternary code length n in A340131) is a(n). Example: numbers 29, 33, 44, 45 and 50 have a ternary length of 4 (see A340131), respectively a(4) = 50.

Ternary code for a(n) is 12..12 for even n and 12..120 for odd n.

			A001006(2) = 2, so a(2) = A340131(2) = 5.
A001006(3) = 4, so a(3) = A340131(4) = 15, etc.

Andrew Howroyd, Table of n, a(n) for n = 2..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Subsequence of A340131.

Vec(5/(1 - 3*x - x^2 + 3*x^3) + O(x^30)) \\ Andrew Howroyd, Jan 08 2021

a(n) = 5*3^(n-2*k)*(9^k-1)/8 where k = floor(n/2).
a(n+1) = 3*a(n) for even n >= 2; a(n+1) = 3*a(n)+5 for odd n >= 3.
a(n) = 5*A033113(n-1).
a(n) = (5/8)*(3^n - (-1)^(n-1) - 2).
a(n) = 2*a(n-1) + 3*a(n-2) + 5 for n > 3.
From Stefano Spezia, Jan 06 2021: (Start)
G.f.: 5*x^2/(1 - 3*x - x^2 + 3*x^3).
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 4. (End)

A348189 Pseudo-involutory Riordan companion of 1 + 2xM(x), where M(x) is the g.f. of A001006.

Original entry on oeis.org

1, 0, 0, 2, 0, 6, 8, 24, 60, 148, 396, 1026, 2744, 7350, 19872, 54102, 148104, 407682, 1127328, 3130542, 8726256, 24407634, 68482776, 192698124, 543642476, 1537443024, 4357677516, 12376868254, 35221087656, 100409367690, 286730523104, 820078634232, 2348966799132

Offset: 1

Alexander Burstein, Oct 06 2021

nonn

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Cf. A001006, A007971, A086246.

a[n_] := SeriesCoefficient[(1 - Sqrt[1-2*x-3*x^2])/(x * (2 + x - Sqrt[1-2*x-3*x^2])), {x, 0, n}]; Array[a, 33, 0] (* Amiram Eldar, Oct 06 2021 *)

my(x='x+O('x^35)); Vec((1-sqrt(1-2*x-3*x^2))/(x*(2+x-sqrt(1-2*x-3*x^2)))) \\ Michel Marcus, Oct 06 2021

G.f.: A(x) = (1 - sqrt(1 - 2*x - 3*x^2))/(x*(2 + x - sqrt(1 - 2*x - 3*x^2))).
If M(x) is the g.f. of A001006, then A(x) = (1 + 2*x*M(x))/(1 + 2*x + 2*x^2*M(x)).
Let M(x) be the g.f. of A001006 and F(x) = 1 + 2*x*M(x) (equivalently, x*F(x) = g.f. of A007971). Then F(-x*A(x)) = 1/F(x).
A(-x*A(x)) = 1/A(x).
G.f.: Let F(x) be the g.f. of A107264, then A(x) = 1 + 2*x^3*A(x)^2*F(x^2*A(x)). - Alexander Burstein, Feb 14 2022

A354292 Primes p such that for all m, M(m) is not divisible by p^2 where M(n) is the n-th Motzkin number A001006.

Original entry on oeis.org

5, 13, 31, 37, 61, 79, 97, 103

Offset: 1

Michel Marcus, May 23 2022

E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, 27(1):245-288, 2015.
Armin Straub, On congruence schemes for constant terms and their applications, arXiv:2205.09902 [math.NT], 2022. See Theorem 3.3 p. 13.

Cf. A001006, A001248, A354293.

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

A045994 a(0)=1, a(n) = M(n) + Sum_{k=1..n-1} M(k)*a(n-k-1), where M(n) are the Motzkin numbers (A001006).

Original entry on oeis.org

1, 1, 3, 7, 18, 47, 125, 337, 918, 2522, 6977, 19415, 54297, 152507, 429974, 1216297, 3450817, 9816460, 27991422, 79989880, 229034820, 656979399, 1887653560, 5431969355, 15653355151, 45167783715, 130491471940, 377426429199

Offset: 0

N. J. A. Sloane

nonn

Apparently the number of grand Motzkin paths of length n that avoid DD starting at level 1. That is, avoiding either positive to negative or negative to positive crossings of the x axis. - David Scambler, Jul 04 2013

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A005773.

m[0] = 1; m[n_] := m[n] = m[n-1] + Sum[m[k]*m[n-k-2], {k, 0, n-2}]; a[0] = 1; a[n_] := a[n] = m[n] + Sum[m[k]*a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 04 2013 *)

a(n):=sum(sum(k/i*sum(binomial(i,j)*binomial(j,2*j-i-k),j,0,i)*binomial(n-i+k-1,k-1)*(-1)^(n-i),i,k,n),k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

G.f.: 1/(1-x(1+x)*M(x)), where M(x) is the generating function for the Motzkin numbers. a(n) = Sum(Sum(k/i*Sum(binomial(i,j)*binomial(j,2*j-i-k),j,0,i)*binomial(n-i+k-1,k-1)*(-1)^(n-i),i,k,n),k,1,n), n>0. - Vladimir Kruchinin, Sep 10 2010

Conjecture: (n+1)*a(n) + 2*(-2*n-1)*a(n-1) + 2*(-n+3)*a(n-2) + (11*n-19)*a(n-3) + (11*n-27)*a(n-4) + 3*(n-3)*a(n-5) = 0. - R. J. Mathar, Sep 27 2013

A069657 Sum( S(n,k) * M(k-1), k=1..n), where S(n,k) = Stirling numbers of the second kind, M(n) = Motzkin numbers, A001006.

Original entry on oeis.org

0, 1, 2, 6, 24, 115, 628, 3818, 25455, 183968, 1428184, 11824098, 103794727, 961461179, 9360372700, 95448502365, 1016413911387, 11273822075837, 129950445723958, 1553488011957986, 19225242250821071, 245899882175001580, 3245812116097119188, 44155099624566615247

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

Cf. A006531, A001006.

A132939 Concatenate Motzkin numbers (A001006).

Original entry on oeis.org

1, 11, 112, 1124, 11249, 1124921, 112492151, 112492151127, 112492151127323, 112492151127323835, 1124921511273238352188, 11249215112732383521885798, 1124921511273238352188579815511, 112492151127323835218857981551141835

Offset: 0

Omar E. Pol, Sep 12 2007

Motzkin numbers: A001006. Cf. A007908, A019518, A059996.

a(13) corrected by Georg Fischer, Dec 30 2022

A134822 Successive digits of Motzkin numbers A001006(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 12 2007

Cf. A007376, A033308.

A144025 Eigentriangle by rows, A001006(n-k)*A005773(k); 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 2, 2, 5, 9, 4, 4, 5, 13, 21, 9, 8, 10, 13, 35, 51, 21, 18, 20, 26, 35, 96, 127, 51, 42, 45, 52, 70, 96, 267, 323, 127, 102, 105, 117, 140, 192, 267, 750, 835, 323, 254, 255, 273, 315, 384, 534, 750, 2123, 2188, 835, 646, 635, 663, 735, 864, 1068, 1500, 2123, 6046

Offset: 0

Gary W. Adamson, Sep 07 2008

Left border = Motzkin numbers, A001006.
Right border = A005773.
Row sums = A005773 shifted: (1, 2, 5, 13, 35, 96, 267,...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
    1;
    1,   1;
    2,   1,   2;
    4,   2,   2,   5;
    9,   4,   4,   5,  13;
   21,   9,   8,  10,  13,  35;
   51,  21,  18,  20,  26,  35,  96;
  127,  51,  42,  45,  52,  70,  96, 267;
  323, 127, 102, 105, 117, 140, 192, 267, 750;
  835, 323, 254, 255, 273, 315, 384, 534, 750, 2123;
  ...
Row 3 = (4, 2, 2, 5) = termwise product of (4, 2, 1, 1) and the first 4 terms of A005773: (1, 1, 2, 5) = (4*1, 2*1, 1*2, 1*5). (4, 2, 1, 1) = the first 4 terms of A001066, reversed.

Cf. A001066, A005773.

Eigentriangle by rows: T(n,k) = A001006(n-k)*A005773(k); 0<=k<=n.

a(53) corrected by Georg Fischer, Apr 29 2025

A168594 G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers (A001006).

Original entry on oeis.org

1, 1, 3, 6, 20, 70, 302, 1386, 6902, 35862, 194202, 1082642, 6191680, 36141118, 214715244, 1294849186, 7911159522, 48888093910, 305165808290, 1921992409066, 12202404037088, 78031629139246, 502263432618224, 3252160882871210

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 302*x^6 +...
A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A133053:
F(x) = 1 + x + 4*x^2 + 16*x^3 + 81*x^4 + 441*x^5 + 2601*x^6 +...+ A001006(n)^2*x^n +...

Cf. A001006, A133053, A168344 (variant).

{a(n)=if(n==0,1,polcoeff(x/serreverse(x*sum(m=0,n,polcoeff((1/x)*serreverse(x/(1+x+x^2+x^2*O(x^m))), m)^2 *x^m)+x^2*O(x^n)),n))}

G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A133053.

A187537 Riordan array (1, (A000045(x)/x-1) *A001006(A000045(x)/x-1) ).

Original entry on oeis.org

1, 3, 1, 9, 6, 1, 31, 27, 9, 1, 113, 116, 54, 12, 1, 431, 493, 282, 90, 15, 1, 1697, 2098, 1383, 556, 135, 18, 1, 6847, 8975, 6567, 3107, 965, 189, 21, 1, 28161, 38640, 30636, 16376, 6070, 1536, 252, 24, 1

Offset: 1

Vladimir Kruchinin, Mar 11 2011

The column with index 0 of the standard array is not incorporated in this triangle. (It contains a 1 followed by zeros.)
The truncated Fibonacci sequence is A000045(x)/x-1 = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5+ ...
The composition with the Motzkin sequence is A001006(...) = 1 + x + 4*x^2 + 15*x^3 + 58*x^4 + 229*x^5 + ...
Eventually this defines the second component in the definition (A000045(...)/x-1)*A001006(...) = x + 3*x^2 + 9*x^3 + 31*x^4 + 113*x^5 + 431*x^6 + ... as seen in the left column of the array.

			     1,
     3,    1,
     9,    6,    1,
    31,   27,    9,    1,
   113,  116,   54,   12,   1,
   431,  493,  282,   90,  15,   1,
  1697, 2098, 1383,  556, 135,  18,  1,
  6847, 8975, 6567, 3107, 965, 189, 21, 1

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

T(n,m):=m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n);

T(n,m) = m*Sum_{k=m..n} Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i)*Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j)/k, n>0, m<=n.

cp's OEIS Frontend

A340395 a(n) = A340131(A001006(n)).

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A348189 Pseudo-involutory Riordan companion of 1 + 2*x*M(x), where M(x) is the g.f. of A001006.

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A354292 Primes p such that for all m, M(m) is not divisible by p^2 where M(n) is the n-th Motzkin number A001006.

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A045994 a(0)=1, a(n) = M(n) + Sum_{k=1..n-1} M(k)*a(n-k-1), where M(n) are the Motzkin numbers (A001006).

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A069657 Sum( S(n,k) * M(k-1), k=1..n), where S(n,k) = Stirling numbers of the second kind, M(n) = Motzkin numbers, A001006.

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A132939 Concatenate Motzkin numbers (A001006).

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Extensions

A134822 Successive digits of Motzkin numbers A001006(n).

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A144025 Eigentriangle by rows, A001006(n-k)*A005773(k); 0<=k<=n.

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A168594 G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers (A001006).

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A187537 Riordan array (1, (A000045(x)/x-1) *A001006(A000045(x)/x-1) ).

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A348189 Pseudo-involutory Riordan companion of 1 + 2xM(x), where M(x) is the g.f. of A001006.