A145603 a(n) is the number of walks from (0,0) to (0,4) that remain in the upper half-plane y >= 0 using 2*n +2 unit steps either up (U), down (D), left (L) or right (R).

1, 35, 720, 12375, 196625, 3006003, 45048640, 668144880, 9859090500, 145173803500, 2136958387520, 31479019635375, 464342770607625, 6861343701121875, 101583106970400000, 1507019252941540800

Offset: 1

Author

Peter Bala, Oct 15 2008

Keywords

• easy
• nonn
• changed

Comments

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145601 and A145602. This sequence is the central column taken from the triangle A145599, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 4.

Links

• R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Crossrefs

A000891, A145599, A145600, A145601, A145602.

Programs

Maple

with(combinat):
a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1);
seq(a(n),n = 1..19);

Formula

a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1).

G.f.: x*hypergeom([5/2, 3, 3, 7/2], [1, 6, 6], 16*x). - Stefano Spezia, Aug 26 2025