The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #6 Jan 10 2022 11:06:22
%S 1,0,1,1,4,5,17,25,76,125,353,625,1681,3130,8138,15708,39848,78995,
%T 196718,398013,977055,2008700,4875392,10152167,24416134,51374143,
%U 122631570,260255863,617373878,1319669672,3114111005,6697099803,15733296513,34011131016,79596651164,172834605692
%N Number of n-steps skew Dyck paths that come back to the x-axis.
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2201.02518">Skew Dyck paths with catastrophes</a>, arXiv:2201.02518 [math.CO], 2022. See Theorem 1 p. 6.
%F 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)).
%t 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 *)
%o (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)))
%K nonn
%O 0,5
%A _Michel Marcus_, Jan 10 2022