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A002026 Generalized ballot numbers (first differences of Motzkin numbers).
(Formerly M1416 N0554)
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 (list; graph; refs; listen; history; text; internal format)



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, 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, Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.1.

C. Defant and K. 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.

Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.

J. A. Sharp & N. J. A. Sloane, Correspondence, 1977


a(n) = sum(b = 1 to (n+1)/2) [ n choose (2b-1) ][ 2b choose 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


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) 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


Cf. A001006, A026300, A026107.

A diagonal of triangle A020474.

See A244884 for a variant.

Sequence in context: A038508 A105695 A244884 * A026938 A086622 A253831

Adjacent sequences:  A002023 A002024 A002025 * A002027 A002028 A002029




N. J. A. Sloane


Additional comments from Emeric Deutsch, Dec 27 2003



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Last modified October 27 01:02 EDT 2020. Contains 338035 sequences. (Running on oeis4.)