A276418 - OEIS

%I #40 May 18 2018 20:48:56

%S 1,2,2,6,6,4,20,20,16,8,70,70,60,40,16,252,252,224,168,96,32,924,924,

%T 840,672,448,224,64,3432,3432,3168,2640,1920,1152,512,128,12870,12870,

%U 12012,10296,7920,5280,2880,1152,256,48620,48620,45760,40040,32032,22880,14080,7040,2560,512,184756,184756,175032,155584,128128,96096,64064,36608,16896,5632,1024

%N Starting a random walk on Z at 0 triangle T(j,k) gives the number of paths of length 2*j returning to 0 exactly k times.

%C The number of paths of odd length 2*j+1 is the same as the number of even length 2*j (returning to 0 exactly k times).

%H Muniru A Asiru, <a href="/A276418/b276418.txt">Table of n, a(n) for n = 0..1325</a>

%F T(j,k) = (2^k)*C(2*j-k,j-k).

%F T(j,0) = T(j,1) for j>0.

%F T(j,0) = A000984(j).

%F T(j,1) = A000984(j) for j>0.

%F T(j,2) = A128650(j+1).

%F T(j,j) = A000079(j).

%F T(j,j-1) = A057711(j+1) for j>0.

%e Triangle T(j,k) begins:

%e       1

%e       2,     2

%e       6,     6,     4

%e      20,    20,    16,     8

%e      70,    70,    60,    40,    16

%e     252,   252,   224,   168,    96,    32

%e     924,   924,   840,   672,   448,   224,    64

%e    3432,  3432,  3168,  2640,  1920,  1152,   512,  128

%e   12870, 12870, 12012, 10296,  7920,  5280,  2880, 1152,  256

%e   48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512

%o (GAP) Flat(List([0..10],j->List([0..j],k->2^k*Binomial(2*j-k,j-k)))); # _Muniru A Asiru_, May 18 2018

%K nonn,tabl,walk,easy

%O 0,2

%A _Franz Vrabec_, Sep 27 2016