%I #22 May 29 2023 11:32:48
%S 1,1,6,42,326,2706,23526,211546,1951494,18366882,175674054,1702686090,
%T 16686795846,165079509042,1646340228006,16534463822010,
%U 167081444125702,1697551974416706,17330661859937670,177699201786231530
%N Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis), where each step is (2,1),(1,2) or (1,-1) and start with (1,2).
%F G.f.: A(x) = 1 + x*D(x)^3, where D(x) is the g.f. of A027307. Also: difference of A027307 and A032349. [Changed formula to include a(0) = 1. - _Paul D. Hanna_, May 28 2023]
%F D-finite with recurrence +n*(2*n+1)*a(n) +(-32*n^2+47*n-17)*a(n-1) +2*(55*n^2-223*n+228)*a(n-2) +3*(-4*n^2+33*n-70)*a(n-3) -(2*n-7)*(n-5)*a(n-4)=0. - _R. J. Mathar_, Jul 24 2022
%F From _Paul D. Hanna_, May 28 2023: (Start)
%F G.f. A(x) = (1/x) * Series_Reversion( x / C(x*C(x)^3) ), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F G.f. A(x) = B(x*A(x)) where B(x) = A(x/B(x)) = C(x*C(x)^3) is the g.f. of A363308, and C(x) is the g.f. of the Catalan numbers (A000108). (End)
%e G.f. A(x) = 1 + x + 6*x^2 + 42*x^3 + 326*x^4 + 2706*x^5 + 23526*x^6 + 211546*x^7 + 1951494*x^8 + 18366882*x^9 + 175674054*x^10 + ...
%o (PARI) /* G.f. A(x) = (1/x)*Series_Reversion( x/C(x*C(x)^3) ) */
%o {a(n) = my(C = (1 - sqrt(1 - 4*x +x^2*O(x^n)))/(2*x)); polcoeff( (1/x)*serreverse(x/subst(C,x,x*C^3)), n)}
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, May 28 2023
%Y Cf. A027307, A032349, A363308.
%K nonn
%O 0,3
%A _Emeric Deutsch_