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, 20511953010895707900, 299127910705042547700, 4379986989467851320000

Offset

1,2

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.

Links

Table of n, a(n) for n=1..20.

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

Formula

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

a(n) ~ 3 * 2^(4*n+1) / (Pi * n^2). - Amiram Eldar, Oct 08 2025

Example

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.

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);

Mathematica

a[n_] := 3 * Binomial[2*n+1, n+2] * Binomial[2*n+1, n-1] / (2*n+1); Array[a, 20] (* Amiram Eldar, Oct 08 2025 *)

Crossrefs

Cf. A000891, A145597, A145600, A145602, A145603.

Sequence in context: A016207 A286347 A016147 * A004344 A038339 A051545

Adjacent sequences: A145598 A145599 A145600 * A145602 A145603 A145604

Keyword

easy,nonn

Author

Peter Bala, Oct 15 2008

Status

approved