A145600 - OEIS

A145600

a(n) is the number of walks from (0,0) to (0,1) 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).

%I #10 Dec 01 2020 07:38:24

%S 1,8,75,784,8820,104544,1288287,16359200,212751396,2821056160,

%T 38013731756,519227905728,7174705330000,100136810390400,

%U 1409850293610375,20002637245262400,285732116760449700

%N a(n) is the number of walks from (0,0) to (0,1) 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).

%C Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145601, A145602 and A145603. This sequence is the central column taken from triangle A145596, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 1.

%D M. Dukes and Y. Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013, Pages 816-842. - From _N. J. A. Sloane_, Feb 21 2013

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

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

%F a(n) = A135389(n-1)/(n+1). - _R. J. Mathar_, Jul 14 2013

%F D-finite with recurrence (n+1)^2*a(n) -4*n*(5*n-1)*a(n-1) +16*(2*n-3)^2*a(n-2)=0. - _R. J. Mathar_, Jul 14 2013

%e a(2) = 8: the 8 walks from (0,0) to (0,1) of three steps are

%e UDU, UUD, URL, ULR, RLU, LRU, RUL and LUR.

%p a(n) := 1/n*binomial(2*n,n+1)*binomial(2*n,n-1);

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

%Y Cf. A000891, A145596, A145601, A145602, A145603.

%K easy,nonn

%O 1,2

%A _Peter Bala_, Oct 14 2008