A097894 Partial sums of A014531.

1, 4, 14, 44, 134, 400, 1184, 3488, 10253, 30108, 88386, 259492, 762085, 2239120, 6582280, 19360432, 56976859, 167774428, 494301778, 1457104948, 4297477252, 12680944960, 37436553544, 110569987344, 326713395019, 965775778420

Offset

1,2

Comments

a(n) = number of peaks at even height in all Motzkin paths of length n+3. Example: a(2)=4 because in the 21 Motzkin paths of length 5 we have altogether 4 peaks at even height (shown between parentheses): HU(UD)D, U(UD)DH, U(UD)HD, UH(UD)D.

This is a kind of Motzkin transform of A121262 because the substitution x -> x*A001006(x) in the independent variable of the g.f. A121262(x) defines a sequence which is 1,0,0,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.

Formula

G.f.: (1-2*x-x^2)/(2*x^3*(1-x)*sqrt(1-2*x-3*x^2))-1/(2*x^3). D-finite with recurrence -(n-1)*(n+3)*a(n) +(n+2)*(3n-1)*a(n-1) +(n-1)*(n+1)*a(n-2) -3*n*(n+1)*a(n-3)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ 3^(n+5/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 01 2014

Maple

ser:=series((1-2*z-z^2)/2/z^3/(1-z)/sqrt(1-2*z-3*z^2)-1/2/z^3, z=0, 32): seq(coeff(ser, z^n), n=1..28);

Mathematica

CoefficientList[Series[((1-2*x-x^2)/(2*x^3*(1-x)*Sqrt[1-2*x-3*x^2])-1/(2*x^3))/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *) (* adapted to the offset by Vincenzo Librandi, Feb 13 2014 *)

Prog

(PARI) x='x+O('x^30); Vec((1-2*x-x^2)/(2*x^3*(1-x)*sqrt(1-2*x-3*x^2))-1/(2*x^3)) \\ G. C. Greubel, Dec 20 2017

Crossrefs

Cf. A014531.

Sequence in context: A000300 A005323 A027831 * A065835 A198643 A097137

Adjacent sequences: A097891 A097892 A097893 * A097895 A097896 A097897

Keyword

nonn

Author

Emeric Deutsch, Sep 03 2004

Status

approved