cp's OEIS Frontend

A276418 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.

%I A276418 #40 May 18 2018 20:48:56
%S A276418 1,2,2,6,6,4,20,20,16,8,70,70,60,40,16,252,252,224,168,96,32,924,924,
%T A276418 840,672,448,224,64,3432,3432,3168,2640,1920,1152,512,128,12870,12870,
%U A276418 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 A276418 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 A276418 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 A276418 Muniru A Asiru, <a href="/A276418/b276418.txt">Table of n, a(n) for n = 0..1325</a>
%F A276418 T(j,k) = (2^k)*C(2*j-k,j-k).
%F A276418 T(j,0) = T(j,1) for j>0.
%F A276418 T(j,0) = A000984(j).
%F A276418 T(j,1) = A000984(j) for j>0.
%F A276418 T(j,2) = A128650(j+1).
%F A276418 T(j,j) = A000079(j).
%F A276418 T(j,j-1) = A057711(j+1) for j>0.
%e A276418 Triangle T(j,k) begins:
%e A276418       1
%e A276418       2,     2
%e A276418       6,     6,     4
%e A276418      20,    20,    16,     8
%e A276418      70,    70,    60,    40,    16
%e A276418     252,   252,   224,   168,    96,    32
%e A276418     924,   924,   840,   672,   448,   224,    64
%e A276418    3432,  3432,  3168,  2640,  1920,  1152,   512,  128
%e A276418   12870, 12870, 12012, 10296,  7920,  5280,  2880, 1152,  256
%e A276418   48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512
%o A276418 (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 A276418 nonn,tabl,walk,easy
%O A276418 0,2
%A A276418 _Franz Vrabec_, Sep 27 2016