A145603 - OEIS

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, 22403455858461298500, 333737751613063999500, 4981648718329849800000, 74506524901378243537500

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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 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

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) = (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

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

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);

MATHEMATICA