%I #19 Apr 11 2023 08:42:05
%S 1,1,3,7,19,56,174,561,1859,6292,21658,75582,266798,950912,3417340,
%T 12369285,45052515,165002460,607283490,2244901890,8331383610,
%U 31030387440,115948830660,434542177290,1632963760974,6151850548776
%N Expansion of (1+x^2*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%C a(n) is the number of lattice paths of n up steps and n down steps that start at the origin with an up step and do not cross the x-axis except possibly at (2n-2,0). - _David Callan_, Mar 14 2004
%C a(n) is the number of parking functions of size n avoiding the patterns 132, 213, 231, and 321. - _Lara Pudwell_, Apr 10 2023
%H Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%F a(n) = A000108(n) + A000108(n-1) (Catalan numbers). - _David Callan_, Mar 14 2004
%F D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-2*n+5)*a(n-2)=0, n>=3 - _R. J. Mathar_, Aug 25 2013
%t Join[{1,1},Total/@Partition[CatalanNumber[Range[30]],2,1]] (* _Harvey P. Dale_, Mar 23 2012 *)
%Y Essentially the same as A005807.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 06 2002