A152171 G.f. := (1-sqrt(1-4*x+4*x^2-4*x^3))/(2(1-x+x^2)x).

1, 1, 1, 2, 5, 12, 29, 73, 190, 505, 1363, 3727, 10306, 28771, 80975, 229512, 654545, 1876899, 5408142, 15650939, 45470545, 132573406, 387775229, 1137575084, 3346189045, 9867291817, 29163523978, 86377998093, 256343194011

Offset: 0

Author

Jun Ma (majun(AT)math.sinica.edu.tw), Nov 27 2008

Keywords

• nonn

Comments

a(n) is the number of Dyck paths of length n without the height of peaks 2 (mod 3) and the height of valleys 1 (mod 3)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2028
• Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-4x+4x^2-4x^3])/(2(1-x+x^2)x),{x,0,40}],x] (* Harvey P. Dale, Feb 07 2022 *)

Formula

D-finite with recurrence: (n+1)*a(n) +(-5*n+1)*a(n-1) +9*(n-1)*a(n-2) +12*(-n+2)*a(n-3) +2*(4*n-11)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jan 25 2020

Extensions

Offset corrected. - R. J. Mathar, Jan 25 2020