A244884 -id:A244884 - cp's OEIS frontend

A002026 Generalized ballot numbers (first differences of Motzkin numbers).

0, 1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326, 135015505407, 385719506620, 1103642686382, 3162376205180, 9073807670316, 26068895429376

Offset: 0

N. J. A. Sloane

Number of ordered trees with n+1 edges, having root of degree 2 and nonroot nodes of outdegree at most 2.
Sequence without the initial 0 is the convolution of the sequence of Motzkin numbers (A001006) with itself.
Number of horizontal steps at level zero in all Motzkin paths of length n. Example: a(3)=5 because in the four Motzkin paths of length 3, (HHH), (H)UD, UD(H) and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether five horizontal steps H at level zero (shown in parentheses).
Number of peaks at level 1 in all Motzkin paths of length n+1. Example: a(3)=5 because in the nine Motzkin paths of length 4, HHHH, HH(UD), H(UD)H, HUHD, (UD)HH, (UD)(UD), UHDH, UHHD and UUDD (where H=(1,0), U=(1,1), D=(1,-1)), we have five peaks at level 1 (shown between parentheses).
a(n) = number of Motzkin paths of length n+1 that start with an up step. - David Callan, Jul 19 2004
Could be called a Motzkin transform of A130716 because the g.f. is obtained from the g.f. x*A130716(x)= x(1+x+x^2) (offset changed to 1) by the substitution x -> x*A001006(x) of the independent variable. - R. J. Mathar, Nov 08 2008
For n >= 1, a(n) is the number of length n permutations sorted to the identity by a consecutive-123-avoiding stack followed by a classical-21-avoiding stack. - Kai Zheng, Aug 28 2020

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).

T. D. Noe, Table of n, a(n) for n = 0..500
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.
L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.
J. B. Cosgrave, The Gauss-Factorial Motzkin connection (Maple worksheet, change suffix to .mw)
R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [hep-ph], 2013.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.1.
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Colin Defant and Kai Zheng, Stack-Sorting with Consecutive-Pattern-Avoiding Stacks, arXiv:2008.12297 [math.CO], 2020.
R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Gennady Eremin, Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words, arXiv:1911.01673 [math.CO], 2019. See p. 2.
Gennady Eremin, Generating function for Naturalized Series: The case of Ordered Motzkin Words, arXiv:2002.08067 [math.CO], 2020.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 19, 21.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], 2022; Journal of Physics A: Mathematical and Theoretical 55 012345, (33pp).
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
J. A. Sharp and N. J. A. Sloane, Correspondence, 1977

Cf. A001006, A026300, A026107, row sums of A348840 and of A348869.
A diagonal of triangle A020474.
See A244884 for a variant.

CoefficientList[Series[4x/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,30}],x] (* Harvey P. Dale, Jul 18 2011 *)
a[n_] := n*Hypergeometric2F1[(1-n)/2, 1-n/2, 3, 4]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Aug 13 2012 *)

my(z='z+O('z^66)); concat(0,Vec(4*z/(1-z+sqrt(1-2*z-3*z^2))^2)) \\ Joerg Arndt, Mar 08 2016

a(n) = A001006(n+1) - A001006(n).
a(n) = Sum_{b = 1..(n+1)/2} C(n, 2b-1)*C(2b, b)/(b+1).
Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, Also T(n, n), where T is the array defined in A026105.
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(k, 2i)*A000108(i+1). - Paul Barry, Jul 18 2003
G.f.: 4*z/(1-z+sqrt(1-2*z-3*z^2))^2. - Emeric Deutsch, Dec 27 2003
a(n) = A005043(n+2) - A005043(n). - Paul Barry, Apr 17 2005
D-finite with recurrence: (n+3)*a(n) +(-3*n-4)*a(n-1) +(-n-1)*a(n-2) +3*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 03 2012
a(n) ~ 3^(n+3/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
G.f.: A(z) satisfies z*A(z) = (1-z)*M(z) - 1, where M(z) is the g.f. of A001006. - Gennady Eremin, Feb 09 2021
a(0) = 0, a(1) = 1; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Nov 09 2021
G.f.: x*M(x)^2 where M(x) = (1 - x - sqrt(1 - 2*x - 3*x^2)) / (2*x^2) is the g.f. of the Motzkin numbers A001006. - Peter Bala, Feb 05 2024

Additional comments from Emeric Deutsch, Dec 27 2003

A340131 Numbers whose ternary expansions have the same number of 1's and 2's and, in each prefix (initial fragment), at least as many 1's as 2's.

Original entry on oeis.org

0, 5, 11, 15, 29, 33, 44, 45, 50, 83, 87, 98, 99, 104, 116, 128, 132, 135, 140, 146, 150, 245, 249, 260, 261, 266, 278, 290, 294, 297, 302, 308, 312, 332, 344, 348, 377, 380, 384, 395, 396, 401, 405, 410, 416, 420, 434, 438, 449, 450, 455, 731, 735, 746, 747

Offset: 1

Gennady Eremin, Dec 29 2020

For a nonzero term, the ternary code starts with 1, otherwise the balance of 1's and 2's is broken already in the one-digit prefix. Therefore 7, 19, 21, etc. (see A039001) are not terms.
As another example, for the integer 52 the balance is broken in the three-digit prefix 122 (the entire ternary code is 1221).
Each term with a ternary code of length k corresponds one-to-one to the Motzkin path of length k that starts with an up step. Therefore, the terms can be called digitized Motzkin paths.
The number of terms with a ternary code of length k is equal to A244884(k). Example: five terms 29, 33, 44, 45 and 50 have a ternary length of 4, respectively A244884(4)=5.

			The first terms 0 and 5 are obvious, because the four intermediate ternary codes 1, 2, 10[3], and 11[4] are rejected due to a violation of the balance of 1's and 2's. Next, the successor function S works: for any term x, the next term is S(x).
Iterating over numbers is inefficient; code suffixes (final digits) can be processed faster. The transition from 0 to 12[5] is generalized for terms that are multiples of 9. For example,
S(10200[99]) = 10212[104], S(1122000[1188]) = 1122012[1193], etc.
In this case, the calculation of the subsequent term is reduced to simply replacing the suffix s = 00 with the subsequent suffix s'= 12.
Another common suffix is s = 02..2 = 02^k (twos are repeated at the end of the ternary code). Then the subsequent suffix is s'= 202..2 = 202^(k-1), i.e., within such a suffix, the first two digits are reversed. Here are some examples:
k = 1, S(1002[29]) = 1020[33], the increment is 4*3^0 = 4;
k = 2, S(110022[332]) = 110202[344], the increment is 4*3^1 = 12;
k = 3, S(10110222[2537]) = 10112022[2573], the increment is 4*3^2 = 36;
k = 4, S(111102222[9800]) = 111120222[9908], the increment is 4*3^3 = 108.
There are 5 such group suffixes.

Gennady Eremin, Table of n, a(n) for n = 1..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.

Subsequence of A039001.
Subsequences: A134752, A168607.
Cf. A244884.

is(n) = {my(d = digits(n, 3), v = [0, 0]); for(i = 1, #d, if(d[i] > 0, v[d[i]]++); if(v[1] < v[2], return(0))); v[1] == v[2] } \\ David A. Corneth, Dec 29 2020

def digits(n, b):
  out = []
  while n >= b:
    out.append(n % b)
    n //= b
  return [n] + out[::-1]
def ok(n):
  t = digits(n, 3)
  if t.count(1) != t.count(2): return False
  return all(t[:i].count(1) >= t[:i].count(2) for i in range(1, len(t)))
print([n for n in range(750) if ok(n)]) # Michael S. Branicky, Dec 29 2020

cp's OEIS Frontend

A002026 Generalized ballot numbers (first differences of Motzkin numbers).

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