A145602 - OEIS

A145602

a(n) is the number of walks from (0,0) to (0,3) that remain in the upper half-plane y >= 0 using 2*n +1 unit steps either up (U), down (D), left (L) or right (R).

1, 24, 392, 5760, 81675, 1145144, 16032016, 225059328, 3173688180, 44986664800, 641087516256, 9183622822400, 132211882468575, 1912322889603000, 27781440618420000, 405248874740582400, 5933888308457316900, 87195180448433455200, 1285508376583653924000, 19010302699089199680000

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OFFSET

1,2

COMMENTS

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145601 and A145603. This sequence is the central column taken from the triangle A145598, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 3.

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

a(n) ~ 2^(4*n+5) / (Pi * n^2). - Amiram Eldar, Oct 08 2025

MAPLE

with(combinat):

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

seq(a(n), n = 1..19);

MATHEMATICA

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