A001006 -id:A001006 - cp's OEIS frontend

A275209 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x)-1.

0, 0, 1, 2, 7, 17, 49, 129, 358, 970, 2679, 7364, 20414, 56634, 157877, 441084, 1236496, 3474672, 9790403, 27648486, 78256907, 221951037, 630723367, 1795576937, 5120487946, 14625574662, 41837955145, 119851980508, 343798121799, 987445317761, 2839518519233

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A274934 with Motzkin numbers replacing connected graph counts.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A275210.

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=1..n/2)-
      `if`(n::odd, 0, (t->t*(t-1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 1, n/2}] - If[OddQ[n], 0, Function[t, t*(t - 1)/2][b[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

a(2n+1) = A275210(2n+1).

A275210 Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x)-1.

Original entry on oeis.org

0, 0, 0, 2, 5, 17, 45, 129, 349, 970, 2658, 7364, 20363, 56634, 157750, 441084, 1236173, 3474672, 9789568, 27648486, 78254719, 221951037, 630717569, 1795576937, 5120472435, 14625574662, 41837913310, 119851980508, 343798008165, 987445317761, 2839518208661

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A216785 with Motzkin numbers replacing connected graph counts.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A275209.

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=1..n/2)-
      `if`(n=0 or n::odd, 0, (t->t*(t+1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
a[n_] := a[n] = Sum[b[j]*b[n - j], {j, 1, n/2}] - If[n == 0 || OddQ[n], 0, Function[t, t*(t + 1)/2][b[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

a(2n+1) = A275209(2n+1).

A295132 a(n) = (2/n)Sum_{k=1..n} (2k+1)M(k)^2 where M(k) is the Motzkin number A001006(k).

Original entry on oeis.org

6, 23, 90, 432, 2286, 13176, 80418, 513764, 3400518, 23167311, 161640554, 1150633512, 8332048638, 61232315553, 455830692210, 3432015694314, 26101221114582, 200295455169015, 1549473966622602, 12074304397434552, 94713783502786686, 747454269790900728

Offset: 1

Zhi-Wei Sun, Nov 15 2017

nonn

Sun (2014) conjectures that for any prime p > 3 we have Sum_{k = 0..p-1} M(k)^2 == (2 - 6*p)(p/3) (mod p^2) and Sum_{k = 0..p - 1} k*M(k)^2 == (9*p - 1)(p/3) (mod p^2), where (p/3) is the Legendre symbol.

Sun (2018) proves that a(n) is always an integer.

			a(2) = 23 since (2/2)*Sum_{k=1..2} (2k + 1)*M(k)^2 = (2*1 + 1)*M(1)^2 + (2*2 + 1)*M(2)^2 = 3*1^2 + 5*2^2 = 23.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schroder numbers, J. Number Theory 183(2018), 146-171.
Zhi-Wei Sun, On Motzkin numbers and central trinomial coefficients, arXiv:1801.08905 [math.CO], 2018.

Cf. A001006, A005043, A295112, A295113.

h := k -> (4*k+2)*hypergeom([(1-k)/2,-k/2],[2],4)^2:
a := proc(n) add(simplify(h(k)),k=1..n): if % mod n = 0 then %/n else -1 fi end:
seq(a(n), n=1..25); # Peter Luschny, Nov 16 2017

M[n_] := M[n] = Sum[Binomial[n, 2k] Binomial[2k, k]/(k + 1), {k, 0, n/2}];
a[n_] := a[n] = 2/n * Sum[(2k + 1) M[k]^2, {k, 1, n}];
Table[a[n], {n, 1, 25}]

a(n) = 2*A005043(n+1)*((6+6/n)*A005043(n) + (2+1/n)*A005043(n+1)). - Mark van Hoeij, Nov 10 2022

A354293 a(n) is the least integer m such that A001006(m) is divisible by prime(n)^2 or -1 if no such m exists.

Original entry on oeis.org

3, 4, -1, 23, 21, -1, 188, 65, 1010, 2231, -1, -1, 1326, 389, 1092, 13196, 1450, -1, 40466, 85553, 665, -1, 5139193, 333, -1, 408241, -1, 3072, 6702, 1393, 5832, 935, 1071, 77421, 292187, 775383, 493135, 4185, 1784560, 10632, 7935, 743003, 13418, 64499, 1746798, 12176, 152551

Offset: 1

Michel Marcus, May 23 2022

sign

Chai Wah Wu, Table of n, a(n) for n = 1..52
Armin Straub, On congruence schemes for constant terms and their applications, arXiv:2205.09902 [math.NT], 2022. see Theorem 3.3 p. 13.

Cf. A000040, A001006, A001248, A354292.

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

catalan(n) = binomial(2*n, n)/(n+1);
M(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1));
a(n) = {my(p=prime(n)); if (p>200, error); if (vecsearch([5, 13, 31, 37, 61, 79, 97, 103], p), return (-1)); my(k=1); while (M(k) % p^2, k++); k;};

a(16)-a(38) from Daniel Suteu, May 23 2022

a(39)-a(47) from Chai Wah Wu, May 23 2022

A377659 a(n) = Motzkin(n) - 2^(n - 1 + 0^n) = A001006(n) - A011782(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 19, 63, 195, 579, 1676, 4774, 13463, 37739, 105442, 294188, 820699, 2291243, 6405310, 17937140, 50327731, 141498983, 398666071, 1125566111, 3184339189, 9026625285, 25636264044, 72940663938, 207889060481, 593474349373, 1696848600299, 4858687934567

Offset: 0

Peter Luschny, Nov 28 2024

nonn

These are the Motzkin words of length n - 1 over the alphabet 0, 1, 2,... that contain at least one digit greater than 1. See the Sage program below.

An analogous construction with the Catalan numbers can be found in A125107.

			  N:       0, 1, 2, 3, 4,  5,  6,   7,   8,   9, ...
  A001006: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...
  A011782: 1, 1, 2, 4, 8, 16, 32,  64, 128, 256, ...
  a:       0, 0, 0, 0, 1,  5, 19,  63, 195, 579, ...
.
For n = 5 the 5 Motzkin words of length 4 that have at least one term > 1 are:
  1221, 1211, 1210, 1121, 0121.
For n = 6 the 19 Motzkin words of length 5 that have at least one term > 1 are:
  12321, 12221, 12211, 12210, 12121, 12111, 12110, 12101, 12100, 11221, 11211, 11210, 11121, 10121, 01221, 01211, 01210, 01121, 00121.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A001006, A011782, A125107.

gf := (1 - x - (1-2*x-3*x^2)^(1/2)) / (2*x^2) - (1 - x) / (1 - 2*x):
ser := series(gf, x, 35): seq(coeff(ser, x, n), n = 0..30);
# Alternative:
a := n -> hypergeom([-n/2 + 1/2, -n/2], [2], 4) - 2^(n - 1 + 0^n);
seq(simplify(a(n)), n = 0..29);

A377659[n_] := If[n < 4, 0, HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {2}, 4] - 2^(n - 1)];
Array[A377659, 50, 0] (* Paolo Xausa, Dec 04 2024 *)

from itertools import islice
show = lambda f, n: print(list(islice(f(), n)))
def aGen():
    a, b, n, z = 1, 2, 2, 1
    yield 0
    while True:
        yield b//n - z
        n += 1; z *= 2
        a, b = b, (3*(n-1)*n*a + (2*n-1)*n*b)//((n+1)*(n-1))
show(aGen, 31)

# Generates Motzkin words (for illustration only).
def motzkin_words(n):
     return IntegerListsLex(length=n+1, min_slope=-1, max_slope=1,
                ceiling=[0]+[+oo for i in range(n-1)]+[0])
def MWList(n, show=True):
    c = 0
    for w in motzkin_words(n):
        if any(p > 1 for p in w):
            c += 1
            if show: print(''.join(map(str, w[1:-1])))
    return c
for n in range(8): print(f"[{n}] -> {MWList(n)}")

a(n) = [x^n] (1 - x - (1 - 2*x - 3*x^2)^(1/2)) / (2*x^2) - (1 - x) / (1 - 2*x).

a(n) = hypergeom([-n/2, -n/2 + 1/2], [2], 4) - 2^(n - 1 + 0^n).

A080894 Expansion of the exponential series exp( x M(x) ) = exp( (1-sqrt(1-2x-3x^2))/(2x) ), where M(x) is the ordinary generating series of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 3, 19, 169, 2001, 29371, 516643, 10590609, 248113729, 6541248691, 191719042131, 6185020391353, 217824649952209, 8316522297035499, 342188317852814371, 15095509523107176481, 710794856254145560833

Offset: 0

Emanuele Munarini, Mar 31 2003

Harvey P. Dale, Table of n, a(n) for n = 0..380
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A001006.

#/Sqrt[E]&/@With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-2x-3x^2])/ (2x)],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Oct 26 2011 *)

E.g.f.: exp((1 - x - sqrt(1 - 2*x - 3*x^2))/(2x)).
a(n) = (n-1)!*Sum_{k=1..n} (1/(k-1)!)*Sum_{j=ceiling((n+k)/2)..n} binomial(n,j)*binomial(j,2*j-n-k). - Vladimir Kruchinin, Aug 11 2010
a(n) ~ 3^(n+1/2)*n^(n-1)/(sqrt(2)*exp(n-1)). - Vaclav Kotesovec, Oct 05 2013
Conjecture D-finite with recurrence: +(-2*n+3)*a(n) +(-2*n^3+9*n^2-9*n+1)*a(n-1) +(n-1)*(n-2)*(4*n^2-2*n-3)*a(n-2) +3*(n-1)*(n-3)*(2*n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Jan 24 2020

A086400 (1/p)*(M(p)-1) where p runs through the odd primes and M(p) = A001006(p) is the p-th Motzkin number.

Original entry on oeis.org

1, 4, 18, 527, 3218, 138634, 957857, 49120018, 20473889132, 156766505690, 74324776203270, 4686290685410776, 37541445026997947, 2445882966702428971, 1327937505693018342712, 743757817682170309535791, 6166664829842021655267818, 3568011384191503508502528953, 250506784439047192764704514076

Offset: 1

Benoit Cloitre, Sep 06 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..317

Cf. A001006.

motz[0] = motz[1] = 1; motz[n_] := motz[n] = ((2*n + 1)*motz[n-1] + 3*(n-1)*motz[n-2])/(n+2); Table[(motz[p]-1)/p, {p, Prime[Range[2, 20]]}] (* Amiram Eldar, Apr 20 2025 *)

Data corrected by Amiram Eldar, Apr 20 2025

A086401 a(n) = M(n) (mod n) where M(n) = A001006(n) is the n-th Motzkin number.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 3, 7, 8, 1, 7, 1, 10, 12, 11, 1, 6, 1, 19, 0, 3, 1, 21, 1, 16, 7, 5, 1, 1, 1, 19, 18, 20, 23, 7, 1, 3, 7, 17, 1, 36, 1, 41, 9, 26, 1, 21, 1, 28, 24, 45, 1, 33, 18, 43, 45, 32, 1, 39, 1, 34, 24, 51, 51, 42, 1, 19, 30, 7, 1, 9, 1, 40, 57, 57, 19, 24, 1, 17, 61, 44, 1, 37, 51

Offset: 1

Benoit Cloitre, Sep 06 2003

nonn

If n is an odd prime a(n) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001006, A266969.

motz[0] = motz[1] = 1; motz[n_] := motz[n] = ((2*n + 1)*motz[n-1] + 3*(n-1)*motz[n-2])/(n+2); a[n_] := Mod[motz[n], n]; Array[a, 100] (* Amiram Eldar, Apr 20 2025 *)

a(n) = 0 if and only if n is in A266969. - Amiram Eldar, Apr 20 2025

Data corrected by Amiram Eldar, Apr 20 2025

A092832 Prime Motzkin numbers (see A001006).

Original entry on oeis.org

2, 127, 15511, 953467954114363

Offset: 1

Eric W. Weisstein, Mar 06 2004

Joel A. Henningsen, Sequences Modulo Primes and Finite State Automata, Master's Thesis, University of South Alabama (2019).
Eric Weisstein's World of Mathematics, Motzkin Number

Cf. A001006, A092831.

a[0]=1;a[n_Integer]:=a[n]=a[n-1]+Sum[a[k]*a[n-2-k], {k, 0, n-2}];lst={};Do[If[PrimeQ[p=a[n]], AppendTo[lst, p]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

A144218 Equals product A*B, where A is an infinite lower triangular matrix with A086246 in every column and B is the diagonal matrix with A001006 as diagonal.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 4, 2, 2, 4, 9, 9, 4, 4, 4, 9, 21, 21, 9, 8, 8, 9, 21, 51, 51, 21, 18, 16, 18, 21, 51, 127, 127, 51, 42, 36, 36, 42, 51, 127, 323, 323, 127, 102, 84, 81, 84, 102, 127, 323, 835, 835, 323, 254, 204, 189, 189, 204, 254, 323, 835, 2188

Offset: 0

Gary W. Adamson, Sep 14 2008

Right border is A001006.
Row sums give A001006 without the initial 1.
Left border is A086246 (A001006 with an additional leading 1).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
    1;
    1,   1;
    1,   1,   2;
    2,   1,   2,   4;
    4,   2,   2,   4,   9;
    9,   4,   4,   4,   9,  21;
   21,   9,   8,   8,   9,  21,  51;
   51,  21,  18,  16,  18,  21,  51, 127;
  127,  51,  42,  36,  36,  42,  51, 127, 323;
  323, 127, 102,  84,  81,  84, 102, 127, 323, 835;
  835, 323, 254, 204, 189, 189, 204, 254, 323, 835, 2188;
  ...
Row 4 = (4, 2, 2, 4, 9) = termwise products of (4, 2, 1, 1, 1) and (1, 1, 2, 4, 9) = (4*1, 2*1, 1*2, 1*4, 1*9).

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A001006, A086246.

nmax = 10;
T[0, 0] = T[1, 0] = 1;
T[n_, 0]  := Hypergeometric2F1[3/2, 1-n, 3, 4] // Abs;
T[n_, n_] := Hypergeometric2F1[(1-n)/2, -n/2, 2, 4];
row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];
T[n_, k_] /; 0Jean-François Alcover, Aug 07 2018 *)

Edited by Joerg Arndt, Jan 26 2024

cp's OEIS Frontend

A275209 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x)-1.

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A275210 Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x)-1.

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A295132 a(n) = (2/n)*Sum_{k=1..n} (2k+1)*M(k)^2 where M(k) is the Motzkin number A001006(k).

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A354293 a(n) is the least integer m such that A001006(m) is divisible by prime(n)^2 or -1 if no such m exists.

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A377659 a(n) = Motzkin(n) - 2^(n - 1 + 0^n) = A001006(n) - A011782(n).

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A080894 Expansion of the exponential series exp( x M(x) ) = exp( (1-sqrt(1-2x-3x^2))/(2x) ), where M(x) is the ordinary generating series of the Motzkin numbers A001006.

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A086400 (1/p)*(M(p)-1) where p runs through the odd primes and M(p) = A001006(p) is the p-th Motzkin number.

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A086401 a(n) = M(n) (mod n) where M(n) = A001006(n) is the n-th Motzkin number.

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A295132 a(n) = (2/n)Sum_{k=1..n} (2k+1)M(k)^2 where M(k) is the Motzkin number A001006(k).