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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 03 2004
STATUS
approved