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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

References

• 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).

Links

• 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

Crossrefs

Cf. A001006, A026300, A026107, row sums of A348840 and of A348869.

A diagonal of triangle A020474.

See A244884 for a variant.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

Additional comments from Emeric Deutsch, Dec 27 2003