A145601 a(n) is the number of walks from (0,0) to (0,2) that remain in the upper half-plane y >= 0 using 2*n unit steps either up (U), down (D), left (L) or right (R).

1, 15, 189, 2352, 29700, 382239, 5010005, 66745536, 901995588, 12342120700, 170724392916, 2384209771200, 33577620944400, 476432168185575, 6805332732133125, 97790670976838400, 1412830549632694500

Offset: 1

Author

Peter Bala, Oct 15 2008

Keywords

• easy
• nonn

Comments

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145602 and A145603. This sequence is the central column taken from triangle A145597, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 2.

Examples

			a(2) = 15: the 15 walks from (0,0) to (0,2) of four steps are:
UUUD, UULR, UURL, UUDU, URUL, ULUR, URLU, ULRU,RUUL, LUUR,
RLUU, LRUU, RULU, LURU and UDUU.
		

Links

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

Crossrefs

A000891, A145597, A145600, A145602, A145603.

Programs

Maple

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

Formula

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