A350662 Number of n-steps skew Dyck paths that come back to the x-axis.

1, 0, 1, 1, 4, 5, 17, 25, 76, 125, 353, 625, 1681, 3130, 8138, 15708, 39848, 78995, 196718, 398013, 977055, 2008700, 4875392, 10152167, 24416134, 51374143, 122631570, 260255863, 617373878, 1319669672, 3114111005, 6697099803, 15733296513, 34011131016, 79596651164, 172834605692

Offset

0,5

Links

Table of n, a(n) for n=0..35.

Helmut Prodinger, Skew Dyck paths with catastrophes, arXiv:2201.02518 [math.CO], 2022. See Theorem 1 p. 6.

Formula

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

Mathematica

CoefficientList[Series[((1 - z^2) (3 - z) (1 - z - 2 z^2 - z^3) - (1 - 4 z - 3 z^2 + z^3 + z^4)*Sqrt[1 - 6 z^2 + 5 z^4])/(2 (1 - 2 z^2 - 6 z^3 - 3 z^4 + z^5 + z^6)), {z, 0, 35}], z] (* Michael De Vlieger, Jan 10 2022 *)

Prog

(PARI) my(z='z+O('z^50)); Vec(((1-z^2)*(3-z)*(1-z-2*z^2-z^3)-(1-4*z-3*z^2+z^3+z^4)*sqrt(1-6*z^2+5*z^4))/(2*(1-2*z^2-6*z^3-3*z^4+z^5+z^6)))

Crossrefs

Sequence in context: A152021 A026678 A026869 * A061806 A306161 A119997

Adjacent sequences: A350659 A350660 A350661 * A350663 A350664 A350665

Keyword

nonn

Author

Michel Marcus, Jan 10 2022

Status

approved