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

0,3

Links

Table of n, a(n) for n=0..19.

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)

Example

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 + ...

Prog

(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

Crossrefs

Cf. A027307, A032349, A363308.

Sequence in context: A107266 A142985 A118351 * A364437 A218755 A165314

Adjacent sequences: A033293 A033294 A033295 * A033297 A033298 A033299

Keyword

nonn

Author

Emeric Deutsch

Status

approved