A354540 Number of decorated Dyck paths of length n ending at arbitrary levels.

1, 1, 2, 3, 7, 11, 26, 43, 102, 175, 416, 733, 1745, 3137, 7476, 13651, 32559, 60199, 143672, 268369, 640823, 1207277, 2884008, 5472821, 13078414, 24973213, 59696622, 114609547, 274037261, 528622499, 1264251474, 2449053107

Offset

0,3

Links

Table of n, a(n) for n=0..31.

H. Prodinger, Partial skew Dyck paths -- a kernel method, arXiv:2108.09785 [math.CO], 2021-2022, chapter 3.

Formula

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

D-finite with recurrence 2*(n+1)*a(n) +(-3*n-5)*a(n-1) +8*(-2*n+3)*a(n-2) +(17*n-23)*a(n-3) +2*(17*n-61)*a(n-4) +(-9*n+41)*a(n-5) +20*(-n+6)*a(n-6) +5*(-n+7)*a(n-7)=0.

Maple

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

Crossrefs

Sequence in context: A294451 A005246 A116406 * A112843 A036651 A049454

Adjacent sequences: A354537 A354538 A354539 * A354541 A354542 A354543

Keyword

nonn

Author

R. J. Mathar, Aug 17 2022

Status

approved