A354539 Number of decorated Dyck paths of length n without peaks at level 1 ending at arbitrary levels.

1, 1, 1, 2, 5, 8, 18, 31, 71, 126, 290, 527, 1218, 2253, 5223, 9796, 22763, 43170, 100502, 192347, 448476, 864887, 2019121, 3919162, 9159252, 17877619, 41819003, 82021628, 192015633

Offset: 0

Author

R. J. Mathar, Aug 17 2022

Keywords

• nonn

Links

• H. Prodinger, Skew Dyck paths having no peaks at level 1, JIS 25 (2022) # 22.1.16, section 2.3.

Crossrefs

Cf. A128723 (ending at level 0).

Programs

Maple

g := (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) ;
taylor(%,z=0,30) ;
gfun[seriestolist](%) ;

Formula

G.f.: (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) .

D-finite with recurrence 12*(n+1)*a(n) +3*(-5*n-11)*a(n-1) +5*(-19*n+29)*a(n-2) +14*(5*n-4)*a(n-3) +2*(93*n-356)*a(n-4) +2*(20*n-81)*a(n-5) +2*(-22*n+217)*a(n-6) +2*(-35*n+268)*a(n-7) +2*(-27*n+182)*a(n-8) +5*(-5*n+39)*a(n-9) +5*(-n+9)*a(n-10)=0.