A033296 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).

1, 1, 6, 42, 326, 2706, 23526, 211546, 1951494, 18366882, 175674054, 1702686090, 16686795846, 165079509042, 1646340228006, 16534463822010, 167081444125702, 1697551974416706, 17330661859937670, 177699201786231530

Offset: 0

Author

Emeric Deutsch

Keywords

• nonn

Examples

			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 + ...
		

Crossrefs

Cf. A027307, A032349, A363308.

Programs

PARI

/* G.f. A(x) = (1/x)*Series_Reversion( x/C(x*C(x)^3) ) */
{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)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, May 28 2023

Formula

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]

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

From Paul D. Hanna, May 28 2023: (Start)

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).

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)