A001006 -id:A001006 - cp's OEIS frontend

A153008 Catalan number A000108(n) minus Motzkin number A001006(n).

0, 0, 0, 1, 5, 21, 81, 302, 1107, 4027, 14608, 52988, 192501, 701065, 2560806, 9384273, 34504203, 127288011, 471102318, 1749063906, 6513268401, 24323719461, 91081800417, 341929853235, 1286711419527, 4852902998951, 18341683253676

Offset: 0

Omar E. Pol, Dec 20 2008

Number of Dyck n-paths with at least one UUU. - David Scambler, Sep 17 2012

Cf. A000108, A001006.

A001006 := proc(n) (3/2)^(n+2)*add( 3^(-k)*A000108(k-1)*binomial(k,n+2-k), k=1..n+2) ; end:
A153008 := proc(n) A000108(n)-A001006(n) ;
end:
seq(A153008(n),n=0..30) ; # R. J. Mathar, Jan 22 2009

MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
a[n_] := CatalanNumber[n] - MotzkinNumber[n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 27 2021 *)

a(n) = A000108(n) - A001006(n).

Conjecture: -(n+1)*(n-3)*(n+2)^2*a(n) +3*(n+1)*(2*n^3-n^2-15*n+8)*a(n-1) -(n-1)*(5*n^3-41*n+48)*a(n-2) -6*(n-1)*(n-2)*(2*n-5)*(n+3)*a(n-3)=0, n>=6 - R. J. Mathar, Mar 04 2018

A153787 Largest proper divisor of Motzkin number A001006(n).

Original entry on oeis.org

1, 2, 3, 7, 17, 1, 19, 167, 1094, 2899, 1, 13945, 56817, 155286, 284489, 785593, 3268191, 9099642, 16950673, 47515853, 133587741, 376586805, 1064242599, 1291914643, 12834909238, 36503886401, 1944142787, 19152993059, 1144562017

Offset: 2

Omar E. Pol, Jan 16 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 2..200

Cf. A001006, A032742, A152766, A152982.

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(divisors(M(i))[tau(M(i))-1], i = 2 .. 32); # Emeric Deutsch, Jan 18 2009

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; lpd[n_] := n / FactorInteger[n][[1, 1]]; Table[lpd[mot[n]], {n, 2, 30}] (* Amiram Eldar, Nov 26 2019 *)

Extended by Emeric Deutsch, Jan 18 2009

a(23)-a(27) and a(29)-a(30) corrected by Amiram Eldar, Nov 26 2019

A154349 Sum of proper divisors minus the number of proper divisors of Motzkin number A001006(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 18, 0, 34, 170, 1643, 3603, 0, 25118, 139063, 474559, 284490, 984006, 6536387, 24265729, 18678366, 96214018, 277799290, 1282283434, 2077807072, 1899874612, 19252363859, 44221482383, 1967547352, 29743945396, 1265868622

Offset: 0

Omar E. Pol, Jan 07 2009

nonn

Note that, if a(n) != 0 then Motzkin number A001006(n) is a composite number (A002808), otherwise A001006(n) is a noncomposite number (A008578). See A152770.

Amiram Eldar, Table of n, a(n) for n = 0..201

Cf. A001006, A001065, A002808, A032741, A008578, A152770, A152981, A152982, A152983, A152988, A152990.

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(sigma(M(n))-M(n)-tau(M(n))+1, n = 0 .. 30); # Emeric Deutsch, Jan 12 2009

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; diff[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Table[diff[mot[n]], {n, 0, 30}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = A001065(A001006(n)) - A032741(A001006(n)) = A152770(A001006(n)).

Extended by Emeric Deutsch, Jan 12 2009

A154558 Triangle read by rows: binomial(n-1,k-1)*A001006(k).

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 12, 9, 1, 8, 24, 36, 21, 1, 10, 40, 90, 105, 51, 1, 12, 60, 180, 315, 306, 127, 1, 14, 84, 315, 735, 1071, 889, 323, 1, 16, 112, 504, 1470, 2856, 3556, 2584, 835, 1, 18, 144, 756, 2646, 6426, 10668, 11628, 7515, 2188

Offset: 1

Gary W. Adamson, Jan 11 2009

			First few rows of the triangle =
1;
1, 2;
1, 4, 4;
1, 6, 12, 9;
1, 8, 24, 36, 21;
1, 10, 40, 90, 105, 51;
1, 12, 60, 180, 315, 306, 127;
1, 14, 84, 315, 735, 1071, 889, 323;
1, 16, 112, 504, 1470, 2856, 3556, 2584, 835;
1, 18, 144, 756, 2646, 6426, 10668, 11628, 7515, 2188;
1, 20, 180, 1080, 4410, 12852, 26670, 38760, 37575, 21880, 5798;
...

Cf. A001006, A000245 (row sums), A005843 (column k=2), A046092 (column k=3).

Binomial transform of a diagonalized version of the Motzkin sequence:
(A001006 (1, 2, 4, 9, 21, 51,...) as the main diagonal and the rest zeros.)
T(n,n) = A001006(n).

A154559 Triangle read by rows, A007318 * (A129186 * (A001006 * 0^(n-k))).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 0, 5, 12, 16, 9, 0, 6, 20, 40, 45, 21, 0, 7, 30, 80, 135, 126, 51, 0, 8, 42, 140, 315, 441, 357, 127, 0, 9, 56, 224, 630, 1176, 1428, 1016, 323, 0, 10, 72, 336, 1134, 2646, 4284, 4572, 2907, 835, 0

Offset: 1

Gary W. Adamson, Jan 11 2009

Row sums = the Catalan numbers, A000108, starting with offset 1:

(1, 2, 5, 14, 42,...).

			First few rows of the triangle =
1;
2, 0;
3, 2, 0;
4, 6, 4, 0;
5, 12, 16, 9, 0;
6, 20, 40, 45, 21, 0;
7, 30, 80, 135, 126, 51, 0;
8, 42, 140, 315, 441, 357, 127, 0;
9, 56, 224, 630, 1176, 1428, 1016, 323, 0;
10, 72, 336, 1134, 2646, 4284, 4572, 2907, 835, 0;
11, 90, 480, 1890, 5292, 10710, 15240, 14535, 8350, 2188;
12, 110, 660, 2970, 9702, 23562, 41910, 53295, 45925, 24068, 5798;
...

Cf. A001006, A000108

Triangle read by rows, A007318 * (A129186 * (A001006 * 0^(n-k)))
Binomial transform of a bidiagonal matrix with (1,0,0,0,..,.) as the main
diagonal and A001006 as the subddiagonal starting (1, 2, 4, 9, 21, 51,...).

A174169 A generalized Chebyshev transform of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, -1, -2, 0, 0, -3, 1, 8, 1, 1, 26, 7, -51, -3, 0, -264, -186, 348, -120, -285, 2697, 2871, -2304, 3393, 8029, -25795, -36872, 16108, -60010, -159683, 213795, 413712, -181857, 833779, 2669534, -1272977, -4030235, 3611168, -9145271, -39467427

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform is the (1,3) Somos-4 sequence A174170.

G.f.: (1-x+3x^2-sqrt(1-2x+3x^2-6x^3+9x^4))/(2x^2)=(1/(1+3x))*M(x/(1+3x^2)), M(x) the g.f. of A010006;
a(n) = sum{k=0..floor(n/2), (-3)^k*A001006(n-2k)}.
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(n-1)*a(n-2) +3*(5-2*n)*a(n-3) +9*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012

A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 4, 8, 25, 65, 197, 571, 1753, 5351, 16746, 52626, 167547, 536559, 1732272, 5622960, 18357211, 60205319, 198323708, 655787680, 2176141555, 7244106347, 24185285341, 80960692691, 271685400443, 913784117809, 3079889039230

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform is the (1,8) Somos-4 sequence A097495(n+2).

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A001006.

Table[Sum[Binomial[n - k, k] 2^k * Hypergeometric2F1[(1 - #)/2, -#/2, 2, 4] &[n - 2 k], {k, 0, Floor[n/2]}], {n, 0, 26}] (* Michael De Vlieger, Feb 02 2017, after Peter Luschny at A001006 *)

G.f.: (1-x-2*x^2-sqrt(1-2*x-7*x^2+4*x^3+4*x^4))/(2*x^2) = (1/(1-2*x))*M(x/(1-2*x^2)), M(x) the g.f. of A010006.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * 2^k * A001006(n-2k).
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +7*(1-n)*a(n-2) +2*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012
a(0) = a(1) = 1; a(n) = a(n-1) + 2 * a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2). - Ilya Gutkovskiy, Nov 09 2021
a(n) ~ 17^(1/4) * (3 + sqrt(17))^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Nov 11 2021

A251571 G.f.: M(F(x)) is a power series in x consisting entirely of positive integer coefficients such that M(F(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 19, 27, 39, 55, 79, 113, 160, 228, 322, 455, 641, 902, 1268, 1777, 2490, 3483, 4864, 6791, 9468, 13189, 18358, 25527, 35473, 49248, 68336, 94751, 131288, 181815, 251627, 348051, 481180, 664885, 918285, 1267663, 1749212, 2412635, 3326303, 4584236, 6315428, 8697260

Offset: 0

Paul D. Hanna, Jan 19 2015

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 +...
such that A(x) = M(F(x)),
where F(x) is the g.f. of A251570:
F(x) = x - x^3 - x^4 + x^5 - x^7 - x^8 + x^10 - x^11 - x^13 - x^14 - x^16 - x^17 - x^18 - x^20 - x^22 - x^26 - x^27 - x^28 - x^29 - x^32 - x^33 - x^35 - x^36 - x^39 - x^41 - x^43 - x^44 - x^45 - x^46 - x^47 - x^48 - x^50 +...
and M(x) is the g.f. of the Motzkin numbers:
M(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + 835*x^9 + 2188*x^10 + 5798*x^11 + 15511*x^12 +...

Cf. A251570, A251691.

/* Prints initial N+2 terms: */
N=100;
/* M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of Motzkin numbers: */
{M=1/x*serreverse(x/(1+x+x^2 +x*O(x^(2*N+10)))); M +O(x^21) }
/* Print terms as you build vector A, then print a(n) at the end: */
{A=[1, 0]; print1("1, 0, ");
for(l=1, N, A=concat(A, -3);
for(i=1, 4, A[#A]=A[#A]+1;
V=Vec(subst(M, x, x*truncate(Ser(A)) +O(x^floor(2*#A+1)) ));
if((sign(V[2*#A])+1)/2==1, print1(A[#A], ", "); break)); );
Vec(subst(M,x,x*Ser(A)))}

A309201 a(n) is the smallest divisor of the Motzkin number A001006(n) not already in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 7, 17, 127, 19, 5, 547, 13, 15511, 15, 6, 9, 284489, 57, 1089397, 12, 73, 11, 21, 35, 63, 119, 6417454619, 38, 107, 31, 1483, 497461, 4644523115569, 51, 10, 37, 953467954114363, 1601, 370537, 1063, 1301337253214147, 43, 18, 1951, 520497658389713341

Offset: 1

N. J. A. Sloane, Jul 25 2019

nonn

Is this a permutation of the positive integers? Daniel Suteu's b-file suggests the answer is no, since powers of 2 >= 8 seem to be missing.
In fact Daniel Suteu points out that Eu and Liu (2008) prove that no Motzkin number is a multiple of 8.
Given any monotonically increasing sequence {b(n): n >= 1} of positive integers we can define a sequence {a(n): n >= 1} by setting a(n) to be smallest divisor of b(n) not already in the {a(n)} sequence. The triangular numbers A000217 produce A111273. A000027 is fixed under this transformation.

Daniel Suteu, Table of n, a(n) for n = 1..191
Eu, Sen-Peng & Liu, Shu-Chung & Yeh, Yeong-nan, Catalan and Motzkin numbers modulo 4 and 8, EJC (2008).

Cf. A000027, A000108, A000217, A111273, A309200.

More terms from Daniel Suteu, Jul 25 2019

A338220 Numbers k such that the Motzkin number A001006(k) is divisible by 5.

Original entry on oeis.org

9, 13, 23, 34, 38, 59, 63, 84, 88, 99, 109, 113, 134, 138, 148, 159, 163, 184, 188, 209, 213, 224, 234, 238, 249, 259, 263, 273, 284, 288, 309, 313, 334, 338, 349, 359, 363, 373, 384, 388, 398, 409, 413, 434, 438, 459, 463, 474, 484, 488, 509, 513, 523, 534, 538

Offset: 1

Amiram Eldar, Jan 30 2021

nonn

The asymptotic density of this sequence is 1/10. It is a disjoint union of 4 sequences: numbers of the form (5*i + 1)*5^(2*j) - 2, (5*i + 2)*5^(2*j-1) - 1, (5*i + 3)*5^(2*j-1) - 2, and (5*i + 4)*5^(2*j) - 1, with i>=0 and j>=1, whose asymptotic densities are 1/120, 1/24, 1/24, and 1/120, respectively (Burns, 2016).

			9 is a term since A001006(9) = 835 = 5 * 167 is divisible by 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Cf. A001006.

Similar sequences, indices of Motzkin numbers divisible by m: A081706 (m = 2), A089119 (m = 3).

motz[0] = motz[1] = 1; motz[n_] := motz[n] = ((2*n + 1)*motz[n - 1] + 3*(n - 1)*motz[n - 2])/(n + 2);  Select[Range[0, 500], Divisible[motz[#], 5] &]

cp's OEIS Frontend

A153008 Catalan number A000108(n) minus Motzkin number A001006(n).

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A153787 Largest proper divisor of Motzkin number A001006(n).

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A154349 Sum of proper divisors minus the number of proper divisors of Motzkin number A001006(n).

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A154558 Triangle read by rows: binomial(n-1,k-1)*A001006(k).

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A154559 Triangle read by rows, A007318 * (A129186 * (A001006 * 0^(n-k))).

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A174169 A generalized Chebyshev transform of the Motzkin numbers A001006.

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A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

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A251571 G.f.: M(F(x)) is a power series in x consisting entirely of positive integer coefficients such that M(F(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + x*M(x) + x*M(x)^2 is the g.f. of the Motzkin numbers A001006.

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A309201 a(n) is the smallest divisor of the Motzkin number A001006(n) not already in the sequence.

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A251571 G.f.: M(F(x)) is a power series in x consisting entirely of positive integer coefficients such that M(F(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.