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

0,4

Links

Table of n, a(n) for n=0..28.

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

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.

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](%) ;

Crossrefs

Cf. A128723 (ending at level 0).

Sequence in context: A063675 A000943 A304966 * A152006 A271619 A197211

Adjacent sequences: A354536 A354537 A354538 * A354540 A354541 A354542

Keyword

nonn

Author

R. J. Mathar, Aug 17 2022

Status

approved